Statistics of the number of records for random walks and Lévy flights on a 1D lattice

Consider a discrete time series {x₀, x₁, x₂, ..., x_n} of n entries representing, e.g. the temperature in a city or the depth of a river or the stock price of a company, all these data being taken on, say, a daily basis. Then a record is said to happen on day i if the ith entry x_i is larger than all previous entries x₀, x₁, x₂, ..., x_i−1. In general, record statistics is expected to be of interest in fields where time series are used and where the size of the entries is a relevant parameter. Such fields include meteorology [1, 2], hydrology [3, 4], insurance and risk management, trading [5–7], economics [8, 9], sports [10–14], etc. Another important application concerns current issues of climatology such as global warming where both temperature records [2, 15–19] and rainfall precipitation records [20–22] play an important role in anticipating future climatic conditions. Statistics of record events have also been found relevant in biology [23], in the theory of spin glasses [24, 25], and in models of growing networks [26].

The mathematical theory of records has been an important subject of research since the paper by Chandler in 1952 (see, e.g. [27–32]). Record statistics is now well understood in the case when the entries x_i's are independent and identically distributed (i.i.d.) random variables. However, in most realistic situations the entries of the time series are correlated and the theory in this case is still piecemeal. For weak correlations, with a finite correlation time, one expects the record statistics for a large n series to be asymptotically similar to the uncorrelated case, but this is no longer true when there are strong correlations between the entries. One of the simplest and most natural time series with strongly correlated entries corresponds to the one of the positions of a one-dimensional random walk, for which many results are known in the mathematics literature, in particular in the context of fluctuation and renewal theory, and can be found in the classical book by Feller [33]. In the physics literature, a comprehensive theory of record statistics for random walks with continuous jumps was worked out by Majumdar and Ziff in [34] (see also [35]). Considering a time series defined by

$$\begin{equation}{x}_{i}={x}_{i-1}+{\eta }_{i},\end{equation} \tag{ 1 }$$

with x₀ = 0 and where the η_i's are i.i.d. random variables drawn from a symmetric and continuous jump distribution f(η), they showed that the joint probability $P\left({R}_{n},{\tau }_{1},{\tau }_{2},\dots ,{\tau }_{{R}_{n}}\right)$ of R_n records in n steps, with respective life-times τ_i (1 ⩽ i ⩽ R_n), is completely independent of f(η) for any n, and not just asymptotically for large n. This remarkable result, which is a consequence of the so-called Sparre Andersen theorem [36], includes also Lévy flights where f(η) ∼ 1/|η|^μ+1 for large η, with 0 < μ < 2. From the joint probability $P\left({R}_{n},{\tau }_{1},{\tau }_{2},\dots ,{\tau }_{{R}_{n}}\right)$ one can derive the distribution of the number of records P(R_n) by integrating out the age degrees of freedom ${\tau }_{1},{\tau }_{2},\dots ,{\tau }_{{R}_{n}}$, which thus is also universal for all n, i.e. independent of f(η). In this paper, we will be mainly interested in P(R_n) and in particular, its first moment, E(R_n), the average number of records up to step n. For a symmetric and continuous jump distribution, Majumdar and Ziff found that these quantities are given by the universal formulas, valid for any n,

$$\begin{equation}P\left({R}_{n}=m\right)=\left(\genfrac{}{}{0.0pt}{}{2n-m+1}{n}\right)\enspace {2}^{-2n+m-1},\end{equation} \tag{ 2 }$$

and

$$\begin{equation}E\left({R}_{n}\right)=\left(2n+1\right)\left(\genfrac{}{}{0.0pt}{}{2n}{n}\right)\enspace {2}^{-2n}.\end{equation} \tag{ 3 }$$

Note that the expressions in equations (2) and (3) corresponds to taking R₀ = 1 (the initial position is counted as the first record). In particular, for large n, equation (3) gives

$$\begin{equation}E\left({R}_{n}\right)=\sqrt{\frac{4\enspace n}{\pi }}+O\left(\frac{1}{\sqrt{n}}\right)\quad \left(n\to +\infty \right),\end{equation} \tag{ 4 }$$

independently of the jump distribution f(η). Similarly, in the limit m, n → +∞ with fixed $m/\sqrt{n}$, the record number distribution P(R_n = m) in equation (2) approaches a scaling form,

$$\begin{equation}P\left({R}_{n}=m\right)\simeq \sqrt{\frac{\pi }{4\enspace n}}\enspace g\left(m\enspace \sqrt{\frac{\pi }{4\enspace n}}\right);\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\quad g\left(x\right)=\frac{2}{\pi }\enspace {\mathrm{e}}^{-{x}^{2}/\pi },\quad x{\geqslant}0,\end{equation} \tag{ 5 }$$

where the scaling function g(x) is a half-Gaussian, normalized to unity. As long as f(η) is symmetric and continuous, the scaling form in equation (5) is also universal, i.e. independent of the jump distribution f(η).

Following [34], record statistics for random walks has since been studied in different, more general, settings (for a recent review on both i.i.d. and random walk cases see [37]). This includes, for instance, continuous-time symmetric random walks [38], discrete-time random walks with a constant drift [39, 40], an ensemble of N independent symmetric random walks [6], random walks in the presence of a measurement error and additive instrumental noises [41], active run-and-tumble particles [42] or random walks with correlated jumps [43]. The complete universality for all n, found in [34] for symmetric and continuous f(η), does not hold in some of these examples, e.g. in the case of the discrete-time random walk in the presence of a constant drift [40]. Nevertheless, these generalizations usually concern random walks with continuous jumps and it is natural to ask what happens to record statistics for random walks with discrete jumps, or 'lattice random walks'. The question was briefly brought up at the very end of [34]. There, it was found that in the particular case where the walker jumps by η = ±1 at each time step, with equal probability 1/2, equation (4) is replaced with

$$\begin{equation}\begin{aligned}\hfill E\left({R}_{n}\right)& =\frac{1}{2}\left[\frac{{{\left(-1\right)}^{n+1}{\Gamma}\left(n-1/2\right)}_{2}{F}_{1}\left(\frac{3}{2},-n;\frac{3}{2}-n;-1\right)}{2\sqrt{\pi }\enspace {\Gamma}\left(n+1\right)}+1\right]\hfill \\ \hfill & =\sqrt{\frac{2n}{\pi }}+\frac{1}{2}+O\left(\frac{1}{\sqrt{n}}\right)\quad \left(n\to +\infty \right),\hfill \end{aligned}\end{equation} \tag{ 6 }$$

where ₂F₁ is the hypergeometric function. This is the only known result so far about record statistics for random walks with discrete jumps. It can be seen in equation (6) that while the expected number of records still grows asymptotically as $\sqrt{n}$ for large n, like in the case of a continuous jump distribution in equation (4), the prefactor $\sqrt{2/\pi }$ of the leading term differs from the one in the continuous case by a factor $1/\sqrt{2}$ and the next correction term 1/2 is a constant while in the continuous case it goes to zero as $O\left({n}^{-1/2}\right)$.

In this paper, we consider a generalized lattice random walk model where the walker stays on a lattice, but can jump in one step by an arbitrary number of lattice units. Our model includes the ±1 random walk model as a special case. For this generalized model, we derive the distribution P(R_n) and its first moment E(R_n) explicitly and discuss their asymptotic behavior for large n. Our principal motivation for studying this model is to test how far the universality of record statistics for symmetric and continuous jump distribution found in reference [34] can be pushed, and how the discreteness of the position of the walker may modify the record statistics.

The rest of the paper is organized as follows. In section 2, we define our model and summarize our main results. In section 3, we determine the generating function of P(R_n) by following a renewal approach, we prove the generalized Sparre Andersen theorem (11), and we use the obtained results to derive the universal scaling form of P(R_n) in the large n limit. In section 4, we determine the generating function of the average number of records E(R_n). In the simple case η = ±1, with equal probability 1/2, we check that this generating function and the associated expression for E(R_n) coincide with the results in [34]. Then, we derive the large n asymptotic behavior of E(R_n) in the general case, and we apply the result to random walks with η = 0, ±1, ±2, where explicit expressions of the coefficients can be obtained. A scaling form for the large n behavior of E(R_n) is derived in the general case when the probability of staying in place (η = 0) is close to 1. Section 5 is devoted to the verification of our analytical predictions via numerical simulations. Finally, we conclude in section 6. Some more technical points are relegated to the appendices.

We consider a 1D random walk starting at the origin, x₀ = 0, and evolving according to equation (1) with the jump distribution f(η) of the form

$$\begin{equation}f\left(\eta \right)=\sum _{h\in \mathbb{Z}}P\left(h\right)\enspace \delta \left(\eta -h\alpha \right),\end{equation} \tag{ 7 }$$

where α > 0 is a given lattice spacing and 0 ⩽ P(h) ⩽ 1 represents the probability to jump by h lattice units in one step, where $h\in \mathbb{Z}$. We further assume that P(h) is symmetric: P(h) = P(−h). Thus, if the walker starts at the origin, it stays on the lattice with lattice spacing α at any time. Indeed, the position of the walker after the ith step is x_i = l_iα where ${l}_{i}\in \mathbb{Z}$ evolves according to l_i = l_i−1 + h_i with l₀ = 0 and where the jumps h_i are i.i.d. random variables with values in $\mathbb{Z}$ drawn from the symmetric probability P(h). We assume that the Fourier transform of the jump distribution

$$\begin{equation}\hat{f}\left(k\right)={\int }_{-\infty }^{+\infty }{\mathrm{e}}^{\mathrm{i}k\eta }f\left(\eta \right)\mathrm{d}\eta =\sum _{h\in \mathbb{Z}}{\mathrm{e}}^{\mathrm{i}kh\alpha }P\left(h\right)\end{equation} \tag{ 8 }$$

has the small k behavior $\hat{f}\left(k\right)\sim 1-\vert ak{\vert }^{\mu }$ (k → 0), where a ≳ α is the characteristic length scale of the jumps and 0 < μ ⩽ 2 is the Lévy index. As the discrete Fourier transform of P(h), $\hat{f}\left(k\right)$ is a periodic function of k, unlike the case of continuous jumps where $\hat{f}\left(k\right)$ goes to zero as k → +∞.

Summary of main results

Before entering the details of the calculations, it is useful to summarize our main results.

• We show that the renewal property of random walks that was used to study the record statistics for symmetric and continuous jump distributions in [34], can be extended to the present model, but with a twist. Like in the continuous case, we show that the generating functions of P(R_n) and E(R_n) for lattice random walks are respectively given by
  $$\begin{equation}\sum _{n{\geqslant}0}P\left({R}_{n}=m\right)\enspace {s}^{n}={\left[1-\left(1-s\right)\enspace {Q}_{{\geqslant}0}\left(s\right)\right]}^{m-1}{Q}_{{\geqslant}0}\left(s\right),\end{equation} \tag{ 9 }$$
  and
  $$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}=\frac{1}{{\left(1-s\right)}^{2}{Q}_{{\geqslant}0}\left(s\right)},\end{equation} \tag{ 10 }$$
  where Q_⩾0(s) = ∑_n⩾0q_⩾0(n)sⁿ is the generating function of the survival probability, ${q}_{{\geqslant}0}\left(n\right)=\mathrm{Prob}\left({x}_{1}{\geqslant}0,\enspace {x}_{2}{\geqslant}0,\dots ,\enspace {x}_{n}{\geqslant}0\left\vert \right.{x}_{0}=0\right)$. The main difference is in the expression for Q_⩾0(s). While in the continuous case, ${Q}_{{\geqslant}0}\left(s\right)=1/\sqrt{1-s}$ is universal due the Sparre Andersen theorem [36], we show that for discrete jump processes on a lattice, it is non-universal and given by
  $$\begin{equation}{Q}_{{\geqslant}0}\left(s\right)\equiv \sum _{n{\geqslant}0}{q}_{{\geqslant}0}\left(n\right)\enspace {s}^{n}=\frac{Z\left(s\right)}{\sqrt{1-s}},\end{equation} \tag{ 11 }$$
  where
  $$\begin{equation}Z\left(s\right)=\mathrm{exp}\left(-\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1-s\hat{f}\left(k\right)\right]\mathrm{d}k\right),\end{equation} \tag{ 12 }$$
  depends explicitly on the jump distribution f(η) through its Fourier transform. The result in equation (11) can be derived from a generalized Sparre Andersen theorem [36] valid for arbitrary jump distributions on a line, discrete or continuous. We also prove this result using an alternative method, which in fact provides a relatively simple non-combinatorial proof of this generalized Sparre Andersen theorem. In the special case of a ±1 random walk, $f\left(\eta \right)=\frac{1}{2}\left[\delta \left(\eta -\alpha \right)+\delta \left(\eta +\alpha \right)\right]$, one has $\hat{f}\left(k\right)=\mathrm{cos}\left(\alpha \enspace k\right)$ and Z(s) in equation (12) can be computed explicitly. One finds $Z\left(s\right)=\left[\sqrt{1+s}-\sqrt{1-s}\right]/s$ independent of α, thus recovering the known result [34, 37], as it should be. Our formula for Z(s) in equation (12) generalizes this special case of ±1 random walks to arbitrary discrete jump processes on a lattice.
• Analyzing the general results above for large n, we find that while the expected number of records E(R_n) is non-universal as depending explicitly on f(η), remarkably the asymptotic large n behavior of the distribution of the scaled record number R_n/E(R_n) turns out to be universal, i.e. independent of f(η). More precisely, we show that the distribution P(R_n = m), in the scaling limit m, n → +∞ with m/E(R_n) fixed, takes the scaling form:
  $$\begin{equation}P\left({R}_{n}=m\right)\sim \frac{1}{E\left({R}_{n}\right)}\enspace g\left(\frac{m}{E\left({R}_{n}\right)}\right);\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\quad g\left(x\right)=\frac{2}{\pi }\enspace {\mathrm{e}}^{-{x}^{2}/\pi },\quad x{\geqslant}0.\end{equation} \tag{ 13 }$$
  Moreover, the scaling function g(x) coincides rather surprisingly with the one of the symmetric and continuous case in equation (5). Thus, all the non-universal features are absorbed in E(R_n), while the distribution of the scaled variable R_n/E(R_n) for large n is super-universal, i.e. independent of f(η) for both continuous and discrete jump processes.
• Finally, our explicit results in equations (10)–(12) allow us to analyze the large n asymptotic behavior of E(R_n) exactly. We find that for arbitrary discrete jump processes, E(R_n) still behaves as $\sqrt{n}$ for large n, like in the continuous case, but now the prefactor and the next subleading term are non-universal as depending explicitly on f(η). More specifically, from equations (10) and (11) near s = 1, we find that the large n behavior of E(R_n) keeping the terms surviving the n → +∞ limit only, is given by
  $$\begin{equation}E\left({R}_{n}\right)\sim \frac{2}{Z\left(1\right)}\enspace \sqrt{\frac{n}{\pi }}+{C}_{\mu }\quad \left(n\to +\infty \right),\end{equation} \tag{ 14 }$$
  with
  $$\begin{equation}{C}_{2}=\frac{\alpha {N}_{0}}{2\enspace a\enspace Z\left(1\right)},\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {C}_{0{< }\mu {< }2}=0,\end{equation} \tag{ 15 }$$
  where N₀ ⩾ 1 is the number of zeros of $1-\hat{f}\left(k\right)$ in the interval 0 ⩽ k ⩽ π/α. In section 4.4 we give some simple examples where Z(1) in equations (14) and (15) can be computed explicitly. For instance, in the simplest case where the walker jumps by h = 0, ±1 at each time step, with P(0) = p and P(±1) = (1 − p)/2 (with 0 ⩽ p < 1), equations (14) and (15) yield
  $$\begin{equation}E\left({R}_{n}\right)\sim \sqrt{\frac{2\left(1-p\right)\enspace n}{\pi }}+\frac{1}{2}\quad \left(n\to +\infty \right),\end{equation} \tag{ 16 }$$
  which shows how the large n behavior in equation (6) is modified when the walker is allowed to stay in place with probability p. The result in equation (16) holds in the limit n → +∞ with 0 ⩽ p < 1 fixed. It is also interesting to consider the scaling limit when p → 1, n → +∞ with the product n (1 − p) kept fixed. In this scaling limit, we find that
  $$\begin{equation}E\left({R}_{n}\right)\simeq \mathcal{F}\left(\left(1-p\right)\enspace n\right),\end{equation} \tag{ 17 }$$
  where the scaling function $\mathcal{F}\left(x\right)$ is explicitly given in equation (80). We also show in appendix B how to generalize this result to any discrete jump distributions with Lévy index 0 < μ ⩽ 2.

To compute the distribution P(R_n = m) of the number of records R_n after n steps, it turns out to be useful to exploit the renewal property of random walks [34] (whether the jumps are continuous or discrete). Namely, instead of trying to get the distribution P(R_n = m) of R_n directly (which actually seems impossible due to the nontrivial memory carried by the random walk), the strategy would be first to compute the joint probability P(R_n = m, τ₁, τ₂, ..., τ_m) of R_n and the time intervals τ₁, τ₂, ..., τ_m between successive records (see figure 1). Note that τ_m is the time interval between the last record and the arrival time n which does not necessarily correspond to a record. Then, by summing P(R_n = m, τ₁, τ₂, ..., τ_m) over the τ_i's one can obtain the distribution of records P(R_n = m). In this section, we follow this strategy to determine the generating function of P(R_n = m) with respect to n ⩾ 0 which encodes all the information about P(R_n = m).

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3.1. Generating function of P(R_n)

We define the survival probability q_⩾0(n) after n steps by

$$\begin{equation}{q}_{{\geqslant}0}\left(n\right)=\mathrm{Prob}\left({x}_{1}{\geqslant}0,\enspace {x}_{2}{\geqslant}0,\dots ,\enspace {x}_{n}{\geqslant}0\left\vert {x}_{0}=0\right.\right),\end{equation} \tag{ 18 }$$

for n ⩾ 1 and q_⩾0(0) = 1. The generating function of q_⩾0(n) is

$$\begin{equation}{Q}_{{\geqslant}0}\left(s\right)=\sum _{n{\geqslant}0}{q}_{{\geqslant}0}\left(n\right)\enspace {s}^{n}.\end{equation} \tag{ 19 }$$

Note that, since we are considering symmetric random walks and using the translational invariance of the walk with respect to the initial position, one also has

$$\begin{align}\hfill {q}_{{\geqslant}0}\left(n\right)& =\mathrm{Prob}\left({x}_{1}{\geqslant}{x}_{0},\enspace {x}_{2}{\geqslant}{x}_{0},\dots ,\enspace {x}_{n}{\geqslant}{x}_{0}\left\vert {x}_{0}\right.\right)\hfill \\ \hfill & =\mathrm{Prob}\left({x}_{1}{\leqslant}{x}_{0},\enspace {x}_{2}{\leqslant}{x}_{0},\dots ,\enspace {x}_{n}{\leqslant}{x}_{0}\left\vert {x}_{0}\right.\right),\hfill \end{align} \tag{ 20 }$$

independent of the starting point x₀. We define the first-crossing time of x₀ to be n if the walker strictly crosses x₀ for the first time during the nth step. The first-crossing time probability φ(n) is then given by

$$\begin{equation}\varphi \left(n\right)=\mathrm{Prob}\left({x}_{1}{\leqslant}{x}_{0},\enspace {x}_{2}{\leqslant}{x}_{0},\dots ,\enspace {x}_{n-1}{\leqslant}{x}_{0},\enspace {x}_{n}{ >}{x}_{0}\left\vert {x}_{0}\right.\right),\end{equation} \tag{ 21 }$$

which is also independent of x₀ by translation invariance of the walk. From equations (20) and (21), one has the relation

$$\begin{equation}\varphi \left(n\right)={q}_{{\geqslant}0}\left(n-1\right)-{q}_{{\geqslant}0}\left(n\right),\end{equation} \tag{ 22 }$$

which implies that the generating function of φ(n) can be written in terms of the generating function of q_⩾0(n) as

$$\begin{equation}{\Phi}\left(s\right)\equiv \sum _{n{\geqslant}1}\varphi \left(n\right)\enspace {s}^{n}=1-\left(1-s\right){Q}_{{\geqslant}0}\left(s\right).\end{equation} \tag{ 23 }$$

Now, by decomposing the walk into m − 1 stretches of duration τ_i (1 ⩽ i < m) between the successive records and one final stretch of duration τ_m between the last record and the last step n, and by using the translation invariance with respect to the initial position in each stretch, one finds that the joint probability P(R_n = m, τ₁, τ₂, ..., τ_m) can be written as [34] (see also [37] for further discussion)

$$\begin{equation}P\left({R}_{n}=m,\enspace {\tau }_{1},\enspace {\tau }_{2},\dots ,\enspace {\tau }_{m}\right)=\varphi \left({\tau }_{1}\right)\varphi \left({\tau }_{2}\right)\cdots \varphi \left({\tau }_{m-1}\right){q}_{{\geqslant}0}\left({\tau }_{m}\right)\enspace {\delta }_{n,\enspace \sum _{i=1}^{m}{\tau }_{i}},\end{equation} \tag{ 24 }$$

where the Kronecker delta enforces that the sum of the record ages τ_i equals the total walk duration n. The distribution of records P(R_n = m) is then obtained by summing this expression (24) over the τ_i's, yielding

$$\begin{equation}P\left({R}_{n}=m\right)=\sum _{{\tau }_{1}{\geqslant}1}\cdots \sum _{{\tau }_{m-1}{\geqslant}1}\sum _{{\tau }_{m}{\geqslant}0}\varphi \left({\tau }_{1}\right)\varphi \left({\tau }_{2}\right)\cdots \varphi \left({\tau }_{m-1}\right){q}_{{\geqslant}0}\left({\tau }_{m}\right)\enspace {\delta }_{n,\enspace \sum _{i=1}^{m}{\tau }_{i}}.\end{equation} \tag{ 25 }$$

Taking the generating function of both sides of equation (25) with respect to n ⩾ 0, one finally obtains

$$\begin{equation}\sum _{n{\geqslant}0}P\left({R}_{n}=m\right)\enspace {s}^{n}={\Phi}{\left(s\right)}^{m-1}{Q}_{{\geqslant}0}\left(s\right)={\left[1-\left(1-s\right)\enspace {Q}_{{\geqslant}0}\left(s\right)\right]}^{m-1}{Q}_{{\geqslant}0}\left(s\right),\end{equation} \tag{ 26 }$$

which shows that the full statistics of the number of records is entirely encoded in the generating function of the survival probability, Q_⩾0(s), that we will now determine.

3.2. Generalized Sparre Andersen theorem

In this subsection, we prove the results in equations (11) and (12) mentioned in section 2. Below, first we provide a short proof using the generalized Sparre Andersen theorem, and then we give an alternative derivation without assuming this theorem. Our derivation then constitutes a new proof of the generalized Sparre Andersen theorem.

Derivation using the generalized Sparre Andersen theorem: the generalized Sparre Andersen theorem [36] states that the generating function Q_⩾0(s) of the survival probability is given by the formula, valid for arbitrary jump distributions (discrete or continuous),

$$\begin{equation}{Q}_{{\geqslant}0}\left(s\right)=\sum _{n{\geqslant}0}{q}_{{\geqslant}0}\left(n\right)\enspace {s}^{n}=\mathrm{exp}\left[\sum _{m=1}^{\infty }\frac{{s}^{m}}{m}\enspace \mathrm{Prob}.\left({x}_{m}{\leqslant}0\right)\right].\end{equation} \tag{ 27 }$$

This is a remarkable theorem as it relates the survival probability q_⩾0(n) up to n steps (which depends on the full history of a trajectory and hence is nonlocal in time) to a local quantity in time, namely the probability Prob.(x_m ⩽ 0) that the walker at step m is non-positive. Using the symmetry of the walk, it is clear that Prob.(x_m < 0) = Prob.(x_m > 0). In addition, the total probability at step m must be unity, i.e. 2Prob.(x_m < 0) + Prob.(x_m = 0) = 1. Hence,

$$\begin{equation}\mathrm{Prob}.\left({x}_{m}{\leqslant}0\right)=\mathrm{Prob}.\left({x}_{m}{< }0\right)+\mathrm{Prob}.\left({x}_{m}=0\right)=\frac{1}{2}\left[1+\mathrm{Prob}.\left({x}_{m}=0\right)\right].\end{equation} \tag{ 28 }$$

Substituting (28) on the right-hand side of equation (27) and using ${\sum }_{m=1}^{\infty }{s}^{m}/m=-\mathrm{ln}\left(1-s\right)$ gives

$$\begin{equation}{Q}_{{\geqslant}0}\left(s\right)=\frac{Z\left(s\right)}{\sqrt{1-s}};\quad \mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}\quad Z\left(s\right)=\mathrm{exp}\left[\frac{1}{2}\enspace \sum _{m=1}^{\infty }\frac{{s}^{m}}{m}\enspace \mathrm{Prob}.\left({x}_{m}=0\right)\right].\end{equation} \tag{ 29 }$$

Thus, it remains to compute Prob.(x_m = 0), which can be done easily. For arbitrary discrete jump processes, it is clear that the position of the walker after m steps, starting from the origin, is simply ${x}_{m}={\sum }_{i=1}^{m}{\eta }_{i}$, where η_i's are i.i.d random variables each drawn from f(η). Hence the Fourier transform of Prob.(x_m) is given by $E\left({\mathrm{e}}^{\mathrm{i}k{x}_{m}}\right)={\left[\hat{f}\left(k\right)\right]}^{m}$. Inverting this Fourier transform and putting x_m = 0, one obtains

$$\begin{equation}\mathrm{Prob}.\left({x}_{m}=0\right)=\frac{\alpha }{2\enspace \pi }{\int }_{-\pi /\alpha }^{\pi /\alpha }{\left[\hat{f}\left(k\right)\right]}^{m}\mathrm{d}k,\end{equation} \tag{ 30 }$$

which can be used in the expression (29) of Z(s) to get

$$\begin{equation}Z\left(s\right)=\mathrm{exp}\left[-\frac{\alpha }{2\pi }\enspace {\int }_{0}^{\pi /\alpha }\mathrm{ln}\left(1-s\enspace \hat{f}\left(k\right)\right)\mathrm{d}k\right],\end{equation} \tag{ 31 }$$

where we used $\hat{f}\left(k\right)=\hat{f}\left(-k\right)$. This completes the proof of the results announced in equations (11) and (12), in section 2.

An alternative derivation: here we derive the expression of Q_⩾0(s) in equation (11) with Z(s) given in equation (12) by an alternative method which does not rely on the generalized Sparre Andersen theorem in equation (27). In fact, the theorem in equation (27) was originally proved by Sparre Andersen using combinatorial arguments [36]. Our method outlined below actually provides an alternative algebraic/non-combinatorial derivation of this theorem. We proceed in two steps. First, we write

$$\begin{equation}{q}_{{ >}0}\left(n\right)=\mathrm{Prob.}\left({x}_{1}{ >}0,\enspace {x}_{2}{ >}0,\dots ,\enspace {x}_{n}{ >}0\left\vert {x}_{0}=0,\enspace \right.\right),\end{equation} \tag{ 32 }$$

for n ⩾ 1 and q_>0(0) = 1. The generating function of q_>0(n) is

$$\begin{equation}{Q}_{{ >}0}\left(s\right)=\sum _{n{\geqslant}0}{q}_{{ >}0}\left(n\right)\enspace {s}^{n}.\end{equation} \tag{ 33 }$$

The first step consists in finding the relation between Q_⩾0(s) and Q_>0(s). Let 0 ⩽ n₁ ⩽ n denote the step at which the walk reaches its minimum position for the first time and π(n₁) the probability of n₁. Split the walk into one part from i = 0 to i = n₁ and a second part from i = n₁ to i = n (i is the step number) (see figure 2 for an illustration). Taking the minimum position as a new space origin and inverting the direction of time in the first part, one gets

$$\begin{equation}\begin{aligned}\hfill \pi \left({n}_{1}\right)& ={q}_{{ >}0}\left({n}_{1}\right){q}_{{\geqslant}0}\left(n-{n}_{1}\right)\hfill \\ \hfill & =\sum _{{n}_{2}{\geqslant}0}{q}_{{ >}0}\left({n}_{1}\right){q}_{{\geqslant}0}\left({n}_{2}\right){\delta }_{n,\enspace {n}_{1}+{n}_{2}},\hfill \end{aligned}\end{equation} \tag{ 34 }$$

and ${\sum }_{{n}_{1}{\geqslant}0}\pi \left({n}_{1}\right)=1$ reads

$$\begin{equation}\sum _{{n}_{1}{\geqslant}0}\sum _{{n}_{2}{\geqslant}0}{q}_{{ >}0}\left({n}_{1}\right){q}_{{\geqslant}0}\left({n}_{2}\right){\delta }_{n,\enspace {n}_{1}+{n}_{2}}=1.\end{equation} \tag{ 35 }$$

Taking the generating function of both sides of equation (35) with respect to n ⩾ 0 and using the definitions in equations (19) and (33), one finds the relation

$$\begin{equation}{Q}_{{ >}0}\left(s\right)\enspace {Q}_{{\geqslant}0}\left(s\right)=\frac{1}{1-s}.\end{equation} \tag{ 36 }$$

Note that in the case of a continuous jump distribution, one has q_>0(n) = q_⩾0(n). Thus, Q_>0(s) = Q_⩾0(s) and equation (36) immediately gives the Sparre Andersen result ${Q}_{{\geqslant}0}\left(s\right)=1/\sqrt{1-s}$ used in [34], which is a very simple proof of this result.

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To get the generalized Sparre Andersen theorem for discrete jumps we need to split the walk into three parts, instead of two as we did to get equation (36). Let 0 ⩽ n₁ ⩽ n denote the step at which the walk reaches its minimum position for the first time, n₁ + ℓ, with 0 ⩽ ℓ ⩽ n − n₁, the step at which it reaches its minimum position for the last time, and π(n₁, ℓ) the joint probability of n₁ and ℓ (see figure 3 for an illustration). We write

$$\begin{equation}{b}_{{\geqslant}0}\left(n\right)=\mathrm{Prob.}\left({x}_{1}{\geqslant}0,\enspace {x}_{2}{\geqslant}0,\dots ,\enspace {x}_{n}=0\left\vert {x}_{0}=0,\enspace \right.\right),\end{equation} \tag{ 37 }$$

for n ⩾ 1 and b_⩾0(0) = 1. The generating function of b_⩾0(n) is

$$\begin{equation}{B}_{{\geqslant}0}\left(s\right)=\sum _{n{\geqslant}0}{b}_{{\geqslant}0}\left(n\right)\enspace {s}^{n}.\end{equation} \tag{ 38 }$$

Split the walk into one part from i = 0 to i = n₁, a second part from i = n₁ to i = n₁ + ℓ, and a third part from i = n₁ + ℓ to i = n. Taking the minimum position as a new space origin and inverting the direction of time in the first part, one gets

$$\begin{equation}\begin{aligned}\hfill \pi \left({n}_{1},\ell \right)& ={q}_{{ >}0}\left({n}_{1}\right){b}_{{\geqslant}0}\left(\ell \right){q}_{{ >}0}\left(n-{n}_{1}-\ell \right)\hfill \\ \hfill & =\sum _{{n}_{2}{\geqslant}0}{q}_{{ >}0}\left({n}_{1}\right){b}_{{\geqslant}0}\left(\ell \right){q}_{{ >}0}\left({n}_{2}\right){\delta }_{n,\enspace {n}_{1}+{n}_{2}+\ell },\hfill \end{aligned}\end{equation} \tag{ 39 }$$

and ${\sum }_{{n}_{1}{\geqslant}0}{\sum }_{\ell {\geqslant}0}\pi \left({n}_{1},\ell \right)=1$ yields

$$\begin{equation}\sum _{{n}_{1}{\geqslant}0}\sum _{{n}_{2}{\geqslant}0}\sum _{\ell {\geqslant}0}{q}_{{ >}0}\left({n}_{1}\right){b}_{{\geqslant}0}\left(\ell \right){q}_{{ >}0}\left({n}_{2}\right){\delta }_{n,\enspace {n}_{1}+{n}_{2}+\ell }=1.\end{equation} \tag{ 40 }$$

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Taking as previously the generating function of both sides of equation (40) with respect to n ⩾ 0 and using the definitions in equations (33) and (38), one finds

$$\begin{equation}{Q}_{{ >}0}{\left(s\right)}^{2}\enspace {B}_{{\geqslant}0}\left(s\right)=\frac{1}{1-s}.\end{equation} \tag{ 41 }$$

Defining then $Z\left(s\right)=\sqrt{{B}_{{\geqslant}0}\left(s\right)}$ and using equation (36) on the left-hand side of equation (41), one obtains the generalized Sparre Andersen theorem

$$\begin{equation}{Q}_{{\geqslant}0}\left(s\right)=\frac{Z\left(s\right)}{\sqrt{1-s}}\enspace ;\quad \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\quad Z\left(s\right)=\sqrt{{B}_{{\geqslant}0}\left(s\right)}.\end{equation} \tag{ 42 }$$

Note that for continuous jumps, all the terms in equation (37) vanish (because the probability that the walker arrives exactly at x_n = 0 is zero if the jumps are continuous), and B_⩾0(s) in equation (38) reduces to B_⩾0(s) = b_⩾0(0) = 1 yielding Z(s) = 1, independently of the (continuous) jump distribution, as expected. For discrete jumps, the expression of Z(s) does depend on the jump distribution. A simple derivation of this expression is given in appendix A. We indeed find that Z(s) coincides with equation (31), namely,

$$\begin{equation}Z\left(s\right)=\sqrt{{B}_{{\geqslant}0}\left(s\right)}=\mathrm{exp}\left(-\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1-s\hat{f}\left(k\right)\right]\mathrm{d}k\right).\end{equation} \tag{ 43 }$$

This completes our alternative derivation of the generalized Sparre Andersen theorem. Note that equations (26), (42) and (43) entirely characterize the full statistics of the number of records for both discrete and continuous jumps (with Z(s) = 1 in the latter case).

3.3. Universal scaling form of P(R_n) in the large n limit

Using the generalized Sparre Andersen theorem (42) in equation (26) and inverting the generating function, one obtains the following integral representation of P(R_n),

$$\begin{equation}P\left({R}_{n}=m\right)=\frac{1}{2i\pi }\enspace \oint \frac{{\left[1-Z\left(s\right)\sqrt{1-s}\right]}^{m-1}Z\left(s\right)}{{s}^{n+1}\sqrt{1-s}}\enspace \enspace \mathrm{d}s.\end{equation} \tag{ 44 }$$

We consider the scaling regime defined by n, m → +∞ with $m/\sqrt{n}=O\left(1\right)$. Making the change of variable s = exp(−p/n) on the right-hand side of equation (44) and using the fact that, in the n → +∞ limit, only the vicinity of s = 1 contributes to the s-integral, one finds that at lowest order in the scaling regime, the integral representation (44) yields

$$\begin{align}\hfill P\left({R}_{n}=m\right)& \sim \frac{Z\left(1\right)}{2i\pi \sqrt{n}}\enspace {\int }_{\mathcal{L}}\frac{1}{\sqrt{p}}\enspace \mathrm{exp}\left(-Z\left(1\right)\frac{m}{\sqrt{n}}\sqrt{p}\right){\mathrm{e}}^{p}\mathrm{d}p\hfill \\ \hfill & \quad {\times}\left(n,\enspace m\to +\infty ,\enspace m/\sqrt{n}=O\left(1\right)\right),\hfill \end{align} \tag{ 45 }$$

where $\mathcal{L}$ is a Bromwich contour. Performing the inverse Laplace transform on the right-hand side of equation (45) (see, e.g. equation 29.3.84 in [44]), one obtains

$$\begin{equation}P\left({R}_{n}=m\right)\sim \frac{Z\left(1\right)}{\sqrt{\pi n}}\enspace \mathrm{exp}\left[-{\left(\frac{Z\left(1\right)m}{2\sqrt{n}}\right)}^{2}\right]\quad \left(n,\enspace m\to +\infty ,\enspace m/\sqrt{n}=O\left(1\right)\right).\end{equation} \tag{ 46 }$$

From this expression it is straightforward to compute the average number of records in the scaling regime. One has

$$\begin{align}\hfill E\left({R}_{n}\right)& =\sum _{m{\geqslant}1}mP\left({R}_{n}=m\right)\sim \sum _{m{\geqslant}1}\frac{Z\left(1\right)\enspace m}{\sqrt{\pi n}}\enspace \mathrm{exp}\left[-{\left(\frac{Z\left(1\right)m}{2\sqrt{n}}\right)}^{2}\right]\hfill \\ \hfill & \sim \frac{4}{Z\left(1\right)}\enspace \sqrt{\frac{n}{\pi }}\enspace {\int }_{0}^{+\infty }x\enspace {\mathrm{e}}^{-{x}^{2}}\mathrm{d}x=\frac{2}{Z\left(1\right)}\enspace \sqrt{\frac{n}{\pi }}\quad \left(n\to +\infty \right),\hfill \end{align} \tag{ 47 }$$

where we have made the change of variable $x=Z\left(1\right)\enspace m/\left(2\sqrt{n}\right)$ and replaced the Riemann sum over m with the corresponding integral over x. Rewriting then the right-hand side of equation (46) in terms of E(R_n) in equation (47), one gets the universal scaling form

$$\begin{equation}P\left({R}_{n}=m\right)\sim \frac{1}{E\left({R}_{n}\right)}\enspace g\left(\frac{m}{E\left({R}_{n}\right)}\right)\quad \left(n,\enspace m\to +\infty ,\enspace m/\sqrt{n}=O\left(1\right)\right),\end{equation} \tag{ 48 }$$

with g(x) = (2/π)exp(−x²/π). The full statistics of the number of records in the scaling regime is thus entirely characterized (at lowest order) by a single parameter, E(R_n), which depends on the jump distribution (through Z(1) in equation (47)). The scaling function itself, g(x), is universal, i.e. independent of the jump distribution. For continuous jumps, Z(1) reduces to Z(1) = 1 independently of the (continuous) jump distribution and the scaling form (48) reduces to the one mentioned in [34] (see equation (5)), as it should be.

The next section is devoted to a detailed study of E(R_n). We show in particular that in the large n limit and for μ = 2, there is a constant correction to the leading behavior in equation (47). For 0 < μ < 2 there is no such a correction and the leading term on the right-hand side of equation (47) is the only term surviving the n → +∞ limit.

As we have just seen, the average number of records, E(R_n), is an important quantity since it is the only non-universal parameter that enters in the full distribution of the number of records in the large n limit. In this section, we study E(R_n) in more details, going further than the simple leading asymptotic behavior (47). As we did for P(R_n), our starting point is the generating function of E(R_n) with respect to n ⩾ 0 which encodes all the information about E(R_n).

4.1. Generating function of E(R_n)

Multiplying both sides of equation (26) by m, summing over all m and using the relation ∑_m⩾0mx^m−1 = 1/(1 − x)², one gets

$$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}=\frac{1}{{\left(1-s\right)}^{2}{Q}_{{\geqslant}0}\left(s\right)}.\end{equation} \tag{ 49 }$$

Replacing then Q_⩾0(s) by the generalized Sparre Andersen result (11), one obtains the generating function of E(R_n) as

$$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}=\frac{1}{{\left(1-s\right)}^{3/2}Z\left(s\right)},\end{equation} \tag{ 50 }$$

with Z(s) given in equation (12).

In the case of a continuous jump distribution, Z(s) = 1 and expanding 1/(1 − s)^3/2 in power series of s, one gets the universal expression of E(R_n) in equation (4) (see equation (8) in [34]). Now, we can go further and use the equations (50) and (43) to see how this universal result is affected when the random walk takes place on a lattice instead of on the line.

4.2. Average number of records for a Bernoulli random walk (h_i = ±1)

It turns out that both ∑_n⩾0E(R_n)sⁿ and E(R_n) can be obtained explicitly when the random walk is a symmetric Bernoulli random walk where the walker jumps by h = ±1 at each step, with P(±1) = 1/2. This simple example of lattice random walk was dealt with at the end of [34]. Here, we check that the equations (50) and (43) do give the same results as the ones in [34], as it should be.

As mentioned in section 2, in this case $\hat{f}\left(k\right)$ reduces to $\hat{f}\left(k\right)=\mathrm{cos}\left(\alpha \enspace k\right)$ and the integral on the right-hand side of equation (43) yields the explicit expression

$$\begin{equation}Z\left(s\right)={\left(\frac{2}{1+\sqrt{1-{s}^{2}}}\right)}^{1/2}=\frac{2}{\sqrt{1+s}+\sqrt{1-s}},\end{equation} \tag{ 51 }$$

independent of α. Using equation (51) in equation (50), one gets

$$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}=\frac{\sqrt{1+s}+\sqrt{1-s}}{2{\left(1-s\right)}^{3/2}},\end{equation} \tag{ 52 }$$

which coincides with the equation (12) in [34]. Expanding the right-hand side of (52) in power series of s, one gets

$$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}=\sum _{n{\geqslant}1}\frac{1}{2}\left[\frac{1}{{\Gamma}\left(n+1\right)}\sum _{j=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{j}\right)\frac{{\Gamma}\left(3/2+j\right)}{{\Gamma}\left(3/2+j-n\right)}+1\right]\enspace {s}^{n},\end{equation} \tag{ 53 }$$

and using the relation (see, e.g. equation 15.4.1 in [44])

$$\begin{equation*}\enspace {}_{2}F_{1}\left(\frac{3}{2},-n;\frac{3}{2}-n;-1\right)={\left(-1\right)}^{n+1}\frac{2\sqrt{\pi }}{{\Gamma}\left(n-1/2\right)}\enspace \sum _{j=0}^{n}\left(\genfrac{}{}{0pt}{}{n}{j}\right)\frac{{\Gamma}\left(3/2+j\right)}{{\Gamma}\left(3/2+j-n\right)},\end{equation*}$$

one obtains the expected expression in the first line of equation (6) (see equation (13) in [34]),

$$\begin{equation}E\left({R}_{n}\right)=\frac{1}{2}\left[\frac{{{\left(-1\right)}^{n+1}{\Gamma}\left(n-1/2\right)}_{2}{F}_{1}\left(\frac{3}{2},-n;\frac{3}{2}-n;-1\right)}{2\sqrt{\pi }\enspace {\Gamma}\left(n+1\right)}+1\right].\end{equation} \tag{ 54 }$$

The large n behavior of equation (54) is more easily obtained from the behavior of equation (52) near its dominant singularity at s = 1. One has

$$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}=\frac{1}{\sqrt{2}\enspace {\left(1-s\right)}^{3/2}}+\frac{1}{2\enspace \left(1-s\right)}+O\left(\frac{1}{\sqrt{1-s}}\right)\quad \left(s\to 1\right),\end{equation} \tag{ 55 }$$

which translates into the large n behavior in the second line of equation (6),

$$\begin{equation}E\left({R}_{n}\right)=\sqrt{\frac{2n}{\pi }}+\frac{1}{2}+O\left(\frac{1}{\sqrt{n}}\right)\quad \left(n\to +\infty \right).\end{equation} \tag{ 56 }$$

4.3. Large n behavior of E(R_n) in the general case

An equation like equation (54) is a remarkable result and, in most cases, it is not possible to derive such an explicit expression of E(R_n) for all n. Nevertheless, as we will see in this section, one can always get the large n behavior of E(R_n), for any lattice random walks and Lévy flights with index 0 < μ ⩽ 2.

Again, the large n behavior of E(R_n) is more easily obtained from the behavior of its generating function in equation (50) near its dominant singularity at s = 1. We thus need to determine the behavior of Z(s) near s = 1. To this end, we rewrite equation (43) as

$$\begin{equation}\begin{aligned}\hfill Z\left(s\right)& =Z\left(1\right)\enspace \mathrm{exp}\left(-\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1+\left(1-s\right)\hat{F}\left(k\right)\right]\mathrm{d}k\right)\hfill \\ \hfill & =Z\left(1\right)\enspace \mathrm{exp}\left(-\frac{\alpha {N}_{0}}{2\pi }{\int }_{0}^{K/2}\mathrm{ln}\left[1+\left(1-s\right)\hat{F}\left(k\right)\right]\mathrm{d}k\right),\hfill \end{aligned}\end{equation} \tag{ 57 }$$

with $\hat{F}\left(k\right)=\hat{f}\left(k\right)/\left[1-\hat{f}\left(k\right)\right]$ and N₀ = 2π/(αK), where K is the period of $\hat{f}\left(k\right)$. For s → 1, the correction to Z(s) = Z(1) in equation (57) depends on the behavior of $\hat{F}\left(k\right)$ near the zeros of $1-\hat{f}\left(k\right)$, where $\hat{F}\left(k\right)$ is large. It can be proved that in the interval 0 ⩽ k ⩽ K/2, there is no other zero than the one at k = 0. Thus, in the limit s → 1, the correction to Z(s) = Z(1) depends on the behavior of $\hat{F}\left(k\right)$ near k = 0, only. For μ = 2, we make the change of variable $k=\sqrt{1-s}\enspace q/a$ and letting s → 1, one gets

$$\begin{equation}Z\left(s\right)\sim Z\left(1\right)\enspace \left(1-\frac{\alpha {N}_{0}}{2a}\sqrt{1-s}\right)\quad \left(s\to 1\right),\end{equation} \tag{ 58 }$$

where we have used the small k behavior $\hat{F}\left(k\right)\sim \vert ak{\vert }^{-2}$ and ${\int }_{0}^{+\infty }\mathrm{ln}\left(1+1/{q}^{2}\right)\mathrm{d}q=\pi$. Similarly, for 1 < μ < 2, we make the change of variable k = (1 − s)^1/μ q/a and using the small k behavior $\hat{F}\left(k\right)\sim \vert ak{\vert }^{-\mu }$, one finds

$$\begin{equation}Z\left(s\right)\sim Z\left(1\right)\enspace \left[1-O{\left(1-s\right)}^{1/\mu }\right]\quad \left(s\to 1\right).\end{equation} \tag{ 59 }$$

For 0 < μ < 1, $\hat{F}\left(k\right)$ is integrable at k = 0 and writing $\mathrm{ln}\left[1+\left(1-s\right)\hat{F}\left(k\right)\right]\simeq \left(1-s\right)\hat{F}\left(k\right)$ on the right-hand side of equation (57), one gets

$$\begin{equation}Z\left(s\right)\sim Z\left(1\right)\enspace \left[1-O\left(1-s\right)\right]\quad \left(s\to 1\right).\end{equation} \tag{ 60 }$$

The case μ = 1 needs a slightly more careful treatment. Skipping the details, one finds that the algebraic singularity on the right-hand side of equation (59) changes to a logarithmic singularity, and one has

$$\begin{equation}Z\left(s\right)\sim Z\left(1\right)\enspace \left[1-O\left(1-s\right)\mathrm{ln}\left(\frac{1}{1-s}\right)\right]\quad \left(s\to 1\right).\end{equation} \tag{ 61 }$$

Putting the asymptotic behavior (58) on the right-hand side of equation (50), one gets

$$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}\sim \frac{1}{{\left(1-s\right)}^{3/2}Z\left(1\right)}+\frac{\alpha {N}_{0}}{\left(1-s\right)\enspace 2aZ\left(1\right)}\quad \left(s\to 1\right),\end{equation} \tag{ 62 }$$

which translates into the following large n behavior of E(R_n) for μ = 2,

$$\begin{equation}E\left({R}_{n}\right)\sim \frac{2}{Z\left(1\right)}\enspace \sqrt{\frac{n}{\pi }}+\frac{\alpha {N}_{0}}{2aZ\left(1\right)}\quad \left(n\to +\infty \right).\end{equation} \tag{ 63 }$$

By doing the same with the asymptotic behaviors (59)–(61), one obtains the large n behavior of E(R_n) for 0 < μ < 2. Keeping the terms surviving the n → +∞ limit only, one finds

$$\begin{equation}E\left({R}_{n}\right)\sim \frac{2}{Z\left(1\right)}\enspace \sqrt{\frac{n}{\pi }}\quad \left(n\to +\infty \right).\end{equation} \tag{ 64 }$$

Note that the subleading terms of E(R_n) corresponding to the small corrections on the right-hand side of equations (59)–(61) go to zero when n → +∞. Finally, putting the equations (63) and (64) together, one obtains the large n behavior of E(R_n) as

$$\begin{equation}E\left({R}_{n}\right)\sim \frac{2}{Z\left(1\right)}\enspace \sqrt{\frac{n}{\pi }}+{C}_{\mu }\quad \left(n\to +\infty \right),\end{equation} \tag{ 65 }$$

with

$$\begin{equation}{C}_{2}=\frac{\alpha {N}_{0}}{2aZ\left(1\right)},\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {C}_{0{< }\mu {< }2}=0,\end{equation} \tag{ 66 }$$

valid for any lattice random walks and Lévy flights with index 0 < μ ⩽ 2. Since the equation $\hat{f}\left(k\right)=1$ has no other solution than k = 0 in the interval 0 ⩽ k ⩽ K/2, and since $\hat{f}\left(k\right)$ is an even periodic function with period K, N₀ = 2π/(αK) is equal to the number of zeros of $1-\hat{f}\left(k\right)$ in the interval 0 ⩽ k ⩽ π/α. The presence of N₀ in equation (66) is crucial to ensure the consistency of the results when there are several different representations of the same physical situations. For instance, h = ±1 with α = 2 and h = ±2 with α = 1 correspond to the same random walk. Both have the same a and Z(1), but not the same α. Without N₀ one would get two different values of C₂, which would make no sense. It can be checked that αN₀ = 2 in both cases, yielding the same value of C₂. Similarly, h = ±1 with α = 1 and h = ±2 with α = 1 correspond to a mere rescaling of the same walk, which should not affect the statistics of records. Both have the same α and Z(1), but not the same a. Again, it can be checked that N₀ compensates for the difference (one has ${N}_{0}/a=\sqrt{2}$ in both cases), yielding the same value of C₂.

Note also that in the case of a continuous jump distribution, one has Z(1) = 1 and α = 0, and equation (65) reduces to the large n behavior in equation (4), valid for all (continuous) jump distribution with 0 < μ ⩽ 2, as it should be.

4.4. Application to random walks with h = 0, ±1, ± 2

As an application, we now consider the random walk on $\mathbb{Z}$, α = 1, defined by P(0) = p₀, P(±1) = p₁, P(±2) = p₂, and P(h) = 0 otherwise (h ≠ 0, ±1, ±2), with p₀ + 2p₁ + 2p₂ = 1. Writing p = p₀ and u = p₂/p₁, one has p₁ = (1 − p)/[2(1 + u)] and p₂ = u(1 − p)/[2(1 + u)]. For this random walk, the Fourier transform of the jump distribution, $\hat{f}\left(k\right)$, reads

$$\begin{align}\hfill \hat{f}\left(k\right)& ={p}_{0}+2{p}_{1}\enspace \mathrm{cos}\enspace k+2{p}_{2}\enspace \mathrm{cos}\enspace 2k\hfill \\ \hfill & =1-2{p}_{1}\left(1-\mathrm{cos}\enspace k\right)-2{p}_{2}\left(1-\mathrm{cos}\enspace 2k\right)\hfill \\ \hfill & =1-\left(\frac{1-p}{1+u}\right)\enspace \left[\left(1-\mathrm{cos}\enspace k\right)+u\left(1-\mathrm{cos}\enspace 2k\right)\right]\hfill \\ \hfill & \sim 1-\frac{\left(1+4u\right)\left(1-p\right)}{2\left(1+u\right)}\enspace {k}^{2}\quad \left(k\to 0\right),\hfill \end{align} \tag{ 67 }$$

yielding

$$\begin{equation}a=\sqrt{\frac{\left(1+4u\right)\left(1-p\right)}{2\left(1+u\right)}}.\end{equation} \tag{ 68 }$$

From the first line of equation (67) it is clear that the period of $\hat{f}\left(k\right)$ is K = 2π if p₁ ≠ 0 and K = π if p₁ = 0. Since α = 1, we then have ${N}_{0}=2\pi /\left(\alpha \enspace K\right)=1+{\delta }_{{p}_{1},0}$. The large n behavior of E(R_n) is then given by the equation (65) for μ = 2 with α = 1, ${N}_{0}=1+{\delta }_{{p}_{1},0}$, a given in equation (68), and

$$\begin{equation}Z\left(1\right)=\sqrt{\frac{1+u}{1-p}}\enspace z\left(u\right),\end{equation} \tag{ 69 }$$

where

$$\begin{equation}z\left(u\right)=\mathrm{exp}\left(-\frac{1}{2\pi }{\int }_{0}^{\pi }\mathrm{ln}\left[1-\mathrm{cos}\enspace k+u\left(1-\mathrm{cos}\enspace 2k\right)\right]\mathrm{d}k\right).\end{equation} \tag{ 70 }$$

4.4.1. Large n behaviors of E(R_n) for fixed p and arbitrary p₁ and p₂

It turns out that the function z(u) in equation (70) can be computed explicitly. An easy way to do it is to compute its derivative z'(u) as given by the derivative of the right-hand side of equation (70) with respect to u, and then integrate the result from 0 to u with $z\left(0\right)=\sqrt{2}$. For arbitrary u ⩾ 0, one obtains

$$\begin{equation}z\left(u\right)=\frac{2\sqrt{2}}{1+\sqrt{1+4u}}.\end{equation} \tag{ 71 }$$

Therefore, substituting (69), (71), and ${N}_{0}=1+{\delta }_{{p}_{1},0}$ in equations (65) and (66) one finds

$$\begin{equation}E\left({R}_{n}\right)\sim \sqrt{n\left(1-p\right)}\enspace A\left(u\right)+B\left(u\right)\left(1+{\delta }_{{p}_{1},0}\right)\quad \left(n\to +\infty \right),\end{equation} \tag{ 72 }$$

with

$$\begin{equation}A\left(u\right)=\frac{1+\sqrt{1+4u}}{\sqrt{2\pi \left(1+u\right)}}\qquad B\left(u\right)=\frac{1}{4}\left(1+\frac{1}{\sqrt{1+4u}}\right),\end{equation} \tag{ 73 }$$

where we recall that u = p₂/p₁.

It is clear that for fixed p₀ = p, the statistics of the records for walks with steps h = 0, ±1 (i.e. p₂ = 0) and the one with steps h = 0, ±2 (i.e. p₁ = 0), should be exactly the same, for any value of n. This implies that taking u = 0 or u = +∞ in equation (72) should give exactly the same result. Indeed, one can check from equation (73) that $A\left(0\right)=A\left(+\infty \right)=\sqrt{2/\pi }$ and $B\left(u\right)\left(1+{\delta }_{{p}_{1},0}\right){\vert }_{u=0,\enspace +\infty }=1/2$, as it should be. Note that since B(+∞) = 1/4, the term $\propto {\delta }_{{p}_{1},0}$ is crucial, as mentioned in the discussion below equation (66). It follows in particular that taking u < +∞ arbitrarily large, one gets ${\mathrm{lim}}_{u\to +\infty }\enspace B\left(u\right)\left(1+{\delta }_{{p}_{1},0}\right)\ne B\left(u\right)\left(1+{\delta }_{{p}_{1},0}\right){\vert }_{u=+\infty }$, which suggests that the limit p₁ → 0 is singular. This might indicate that the limits n → ∞ and p₁ → 0 do not commute, since one can check that for any finite n, E(R_n) is a smooth function of p₁. Another interesting consequence of the simple fact that A(0) = A(+∞) is that, since A(u) is a continuous function of u, it admits (at least) one extremum. It turns out that A(u) admits a single maximum at u = 2 for which $A\left(2\right)=\left(2/3\right)\sqrt{6/\pi }$. Note that it is not a priori obvious that (i) the function A(u) is a non-monotonic function of u and (ii) that it has a single maximum.

4.4.2. Scaling form of E(R_n) for large n and n(1 − p) = O(1): p₂ = 0 and general case

We note that the asymptotic expression in equation (72) requires (1 − p)n ≫ 1 to be valid (not just n ≫ 1). In the opposite limit (1 − p)n ≪ 1 corresponding to letting p ≡ P(h = 0) → 1 first, then n → +∞, the walker gets stuck at the initial position x₀ = 0 and there is only one record (the first one at the initial position), yielding E(R_n) = 1. To conclude this section, we derive a uniform scaling form of E(R_n) in the large n limit which describes the crossover between equation (65) for (1 − p)n ≫ 1 and E(R_n) ∼ 1 for (1 − p)n ≪ 1. For simplicity, here we give the details of the calculations for the simple case h = 0, ±1 only. The derivation of the scaling form in the general case is explained in detail in appendix B.

First, we determine the large n behavior of E(R_n) as given by equation (65) for n → +∞ with fixed p < 1. The case h = 0, ±1 corresponds to taking p₂ = 0 (i.e. u = 0) in equation (72). It reads, using $A\left(0\right)=\sqrt{2/\pi }$ and B(0) = 1/2,

$$\begin{equation}E\left({R}_{n}\right)\sim \sqrt{\frac{2\left(1-p\right)\enspace n}{\pi }}+\frac{1}{2}\quad \left(n\to +\infty \right),\end{equation} \tag{ 74 }$$

which shows how the large n behavior in equation (6) is modified when the walker is allowed to stay in place with probability p.

The scaling form of E(R_n) is obtained by considering the limits n → +∞ and p → 1, keeping the scaling variable (1 − p)n fixed. Injecting $\hat{f}\left(k\right)=p+\left(1-p\right)\mathrm{cos}\enspace k$, which corresponds to equation (67) with u = 0, into equation (43), one finds

$$\begin{equation}Z\left(s\right)=\frac{2}{\sqrt{\left(1-s\right)+2s\left(1-p\right)}+\sqrt{1-s}}.\end{equation} \tag{ 75 }$$

Using equation (75) on the right-hand side of equation (50), one obtains

$$\begin{equation}\sum _{n{\geqslant}0}E\left({R}_{n}\right)\enspace {s}^{n}=\frac{\sqrt{\left(1-s\right)+2s\left(1-p\right)}+\sqrt{1-s}}{2{\left(1-s\right)}^{3/2}},\end{equation} \tag{ 76 }$$

from which one gets the following integral representation for E(R_n),

$$\begin{equation}E\left({R}_{n}\right)=\frac{1}{2i\pi }\enspace \oint \frac{\sqrt{\left(1-s\right)+2s\left(1-p\right)}+\sqrt{1-s}}{2{s}^{n+1}{\left(1-s\right)}^{3/2}}\enspace \enspace \mathrm{d}s.\end{equation} \tag{ 77 }$$

Making the change of variable s = exp(−λ/n) in equation (77) and using the fact that, in the n → +∞ limit, only the vicinity of s = 1 contributes to the s-integral, one finds that at lowest order in the scaling regime, the integral representation of E(R_n) reads

$$\begin{equation}E\left({R}_{n}\right)\sim \frac{1}{2i\pi }\enspace {\int }_{\mathcal{L}}\frac{\sqrt{\lambda +2n\left(1-p\right)}+\sqrt{\lambda }}{2{\lambda }^{3/2}}\enspace \enspace {\mathrm{e}}^{\lambda }\mathrm{d}\lambda \quad \left(n\to +\infty ,\enspace p\to 1\right),\end{equation} \tag{ 78 }$$

where $\mathcal{L}$ is a Bromwich contour. Performing then the inverse Laplace transform on the right-hand side of equation (78), one gets the scaling form

$$\begin{equation}E\left({R}_{n}\right)\simeq \mathcal{F}\left[\left(1-p\right)\enspace n\right],\end{equation} \tag{ 79 }$$

valid for n → +∞, p → 1, and fixed (1 − p)n = O(1), with the scaling function

$$\begin{equation}\mathcal{F}\left(x\right)=\left[\left(\frac{1}{2}+x\right)\enspace {I}_{0}\left(x\right)+x\enspace {I}_{1}\left(x\right)\right]{\mathrm{e}}^{-x}+\frac{1}{2},\end{equation} \tag{ 80 }$$

where I_ν(x) is the modified Bessel function of order ν. From the large and small argument behaviors of I_ν(x), one readily obtains

$$\begin{equation}\mathcal{F}\left(x\right)\sim \begin{cases}\sqrt{2x/\pi }+1/2\quad \hfill & \left(x\to +\infty \right),\hfill \\ 1+x/2\quad \hfill & \left(x\to 0\right),\hfill \end{cases}\end{equation} \tag{ 81 }$$

from which it can be checked that the scaling form in equation (79) describes the crossover between equation (74) for (1 − p)n ≫ 1, and E(R_n) ∼ 1 in the opposite limit (1 − p)n ≪ 1. Let us note that a similar scaling limit for E(R_n) was studied recently in reference [22] for time series with i.i.d entries {x_i} in the context of the Bernoulli model of rainfall precipitation records, where a given day is dry with probability p and wet with probability (1 − p), and a record is counted only for a wet day. Due to the presence of strong correlations between the x_i's in the case of random walks with discrete jumps, our scaling function $\mathcal{F}\left(x\right)$ in equation (81) is however quite different from that in the Bernoulli model of i.i.d variables.

It is actually possible to generalize the scaling form in equation (79) to any discrete jump distribution with Lévy index 0 < μ ⩽ 2. The interested reader will find the details of the calculations in appendix B. Writing ${Z}_{0}\left(1\right)=Z\left(1\right)\sqrt{1-p}$ independent of p (for fixed ratios P(h = m)/P(h = 1), m ⩾ 1), one finds

$$\begin{equation}E\left({R}_{n}\right)\sim \mathcal{G}\left[\left(1-p\right)\enspace n\right],\end{equation} \tag{ 82 }$$

valid for n → +∞ and p → 1 with fixed (1 − p)n = O(1), where the scaling function $\mathcal{G}\left(x\right)$ is given by

$$\begin{equation}\mathcal{G}\left(x\right)=\frac{\sqrt{x}}{2i\pi \enspace {Z}_{0}\left(1\right)}\enspace {\int }_{\mathcal{L}}\mathrm{exp}\left(\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1+\frac{\lambda /x}{1-{\hat{f}}_{0}\left(k\right)}\right]\mathrm{d}k\right)\enspace \frac{{\mathrm{e}}^{\lambda }}{{\lambda }^{3/2}}\mathrm{d}\lambda .\end{equation} \tag{ 83 }$$

In equation (83), ${\hat{f}}_{0}\left(k\right)$ denotes the Fourier transform of the jump distribution corresponding to p = 0 with fixed ratios P(h = m)/P(h = 1), m ⩾ 1. (Note that Z₀(1) is nothing but Z(1) in equation (43) with ${\hat{f}}_{0}\left(k\right)$ instead of $\hat{f}\left(k\right)$). It can then be checked from the large and small argument behaviors of $\mathcal{G}\left(x\right)$ given in equations (B7) and (B8) that the scaling form in equation (82) matches smoothly between equation (65) for (1 − p)n ≫ 1 and E(R_n) ∼ 1 for (1 − p)n ≪ 1. This result gives the scaling form of the large n behavior of E(R_n) when P(h = 0) → 1 in the general case of any discrete jump distribution with Lévy index 0 < μ ⩽ 2.

In this section, we present numerical simulations of discrete random walks in the case considered in section 4.4, i.e. α = 1 and h = 0, ±1, ±2, where exact analytical results can be obtained for the average number of records E(R_n) in the limit n ≫ 1 (see equations (72) and (73)). Writing, like in section 4.4, p = P(0), p₁ = P(±1), and p₂ = P(±2), we study three different cases corresponding to three different values of the ratio u = p₂/p₁. Namely, u = 1, u = 1/2, and u = 0. In each case, we have simulated random walks up to n = 5000 steps for two different values of p (the probability of staying in place): p = 0.1 and p = 0.5. The data presented here have been obtained by averaging over 10⁵ independent realizations of the random walk.

In figures 4(a) and (b) we show our data for u = 1 and u = 1/2, respectively. As we can see, the agreement between the numerics and our analytical (asymptotic) results, given in equations (72) and (73), is very good. In figure 5 we show our data for the case u = 0, i.e. p₂ = 0. Figure 5(a) shows a comparison between our data for the average number of records E(R_n) and our asymptotic prediction in equations (72) and (73) as a function of n, with u = 0, for p = 0.1 and p = 0.5. Here again the agreement is excellent.

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Figure 5(b) shows a comparison between our numerical computation of E(R_n) and the analytical values obtained from the generating function for u = 0 and n = O(1). More specifically, using the expression (43) of Z(s) with α = 1 and $\hat{f}\left(k\right)=p+\left(1-p\right)\mathrm{cos}k$ (which corresponds to u = 0) on the right-hand side of equation (50), one gets

$$\begin{equation}\sum _{n=0}^{\infty }E\left({R}_{n}\right){s}^{n}=\frac{1}{{\left(1-s\right)}^{3/2}}\enspace \mathrm{exp}\left[\frac{1}{2\pi }{\int }_{0}^{\pi }\mathrm{ln}\left(1-sp-s\left(1-p\right)\mathrm{cos}k\right)\right],\end{equation} \tag{ 84 }$$

and by expanding the right-hand side in powers of s (which can be done very easily with, e.g. Mathematica) one obtains, for the first values of n,

$$\begin{align}\hfill & E\left({R}_{0}\right)=1,\qquad E\left({R}_{1}\right)=\frac{3}{2}-\frac{p}{2},\qquad E\left({R}_{2}\right)=\frac{1}{4}\left(7-2p-{p}^{2}\right)\hfill \\ \hfill & E\left({R}_{3}\right)=2-\frac{3p}{4}-\frac{{p}^{3}}{4},\qquad E\left({R}_{4}\right)=\frac{1}{16}\left(35-12p-6{p}^{2}+4{p}^{3}-5{p}^{4}\right)\hfill \\ \hfill & E\left({R}_{5}\right)=\frac{1}{16}\left(38-15p-10{p}^{3}+10{p}^{4}-7{p}^{5}\right)\hfill \\ \hfill & E\left({R}_{6}\right)=\frac{1}{32}\left(81-30p-15{p}^{2}+20{p}^{3}-45{p}^{4}+42{p}^{5}-21{p}^{6}\right)\hfill \\ \hfill & E\left({R}_{7}\right)=\frac{1}{32}\left(86-35p-35{p}^{3}+70{p}^{4}-105{p}^{5}+84{p}^{6}-33{p}^{7}\right).\hfill \end{align} \tag{ 85 }$$

In figure 5(b) we compare these exact analytical values with our numerical simulations for p = 0.1 and p = 0.5. The comparison shows a perfect agreement.

Finally, in figure 6, we show plots of E(R_n) for three different values of p close to one, namely p = 0.9, p = 0.95, and p = 0.99, which confirm the scaling form predicted in equation (79) together with the expression of the scaling function $\mathcal{F}\left(x\right)$ given in equation (80)—see figure 6(b)).

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In this paper we have studied a general question concerning the universality of the statistics of the number of records R_n for a discrete-time series whose entries represent the successive positions of an n-step random walk in one dimension. Starting at the origin, at each discrete-time step the walker jumps by a certain distance drawn from a symmetric distribution f(η). If the jump distribution f(η) is continuous, it was known [34] that the statistics of records is strongly universal, i.e. independent of f(η) for all n. What happens to this universality if the walk takes place on a discrete lattice, i.e. if f(η) is nonzero for integer values of η only, was still largely unknown. For the special case of nearest neighbor lattice walk when η = ±1, previous exact results [34] (see also [37]) show that while the average number of records E(R_n) still grows asymptotically as $\sqrt{n}$ for large n, like in the continuous case, the prefactor of this square root growth as well as the next subleading term differ from the ones in the continuous case. This raises the natural question of whether the results from ±1 walk can be generalized to more general jump processes on a 1D lattice, where the walker may jump an arbitrary number of lattice units in one step, without being necessarily restricted to nearest neighbor walk. This question about the record statistics for general discrete jump processes may have relevance, for instance, when the entries of the time series represent the price of a stock on different days. The day-to-day jump in the stock price is often rounded to the nearest integer and stock price effectively moves on a lattice. The purpose of this paper was to derive the record statistics for such a general discrete jump process on a 1D lattice and investigate whether there exist universal results at all, or perhaps the universality holds only asymptotically for large n.

Our main conclusion is that the strong universality, valid for arbitrary n, in the case of continuous jump distributions, no longer holds for discrete jump processes. However, a vestige of universality still remains asymptotically for large n. We found that for arbitrary symmetric discrete jump processes, the distribution of the scaled record number R_n/E(R_n) converges, for large n, to a universal half-Gaussian distribution, like in the case of continuous jumps. The dependence on the jump distribution f(η) is entirely absorbed in the expected number of records E(R_n), leaving the scaled distribution itself independent of f(η). By computing the large n behavior of E(R_n) for general discrete jump processes, we found that while it still grows as $\sqrt{n}$ for large n irrespective of f(η), the prefactor of $\sqrt{n}$ as well as the next subleading term depend explicitly on the jump distribution.

A technical bonus of our work is an alternative derivation of the celebrated generalized Sparre Andersen theorem that was originally derived using combinatorial arguments [36]. Our derivation relies more on an algebraic and non-combinatorial approach. Finally, the techniques developed in this paper will hopefully be useful to study other observables for discrete jump processes on a lattice, such as the statistics of the maximum displacement, the time at which the maximum/minimum occurs etc.

In this paper, we have restricted ourselves to the study of record statistics for symmetric lattice random walks. A natural question is to ask how this record statistics will be affected by the presence of a nonzero drift that would make the jump distribution on the lattice asymmetric. Indeed, for continuous jump distribution (not necessarily symmetric), the record statistics has been studied using the generalized Sparre Andersen theorem. In this case, even though the full universality of the record statistics at all step n no longer holds for asymmetric jumps, there remains a vestige of universality in the limit of large n [40]. It would be interesting to see how the drift affects the record statistics in the case of lattice random walks studied in this paper.

In this appendix, we derive the expression of the function Z(s) given in equation (43). We write

$$\begin{equation}{b}_{{ >}0}\left(n\right)=\mathrm{Prob.}\left({x}_{1}{ >}0,\enspace {x}_{2}{ >}0,\dots ,\enspace {x}_{n}=0\left\vert {x}_{0}=0\right.\right),\end{equation} \tag{ A1 }$$

for n ⩾ 1 and b_>0(0) = 1. The generating function of b_>0(n) is

$$\begin{equation}{B}_{{ >}0}\left(s\right)=\sum _{n{\geqslant}0}{b}_{{ >}0}\left(n\right)\enspace {s}^{n}.\end{equation} \tag{ A2 }$$

Similar expressions for b_⩾0(n) and B_⩾0(s) are given in equations (37) and (38), respectively. For n ⩾ 1, one has the relation

$$\begin{equation}{b}_{{\geqslant}0}\left(n\right)=\sum _{m=0}^{n-1}{b}_{{\geqslant}0}\left(m\right){b}_{{ >}0}\left(n-m\right),\end{equation} \tag{ A3 }$$

which is easily seen by taking for m the last time at which the walker touches x = 0 before arriving at x_n = 0. Writing

$$\begin{equation*}\sum _{m=0}^{n-1}{b}_{{\geqslant}0}\left(m\right){b}_{{ >}0}\left(n-m\right)=\sum _{{m}_{1}{\geqslant}0}\sum _{{m}_{2}{\geqslant}1}{b}_{{\geqslant}0}\left({m}_{1}\right){b}_{{ >}0}\left({m}_{2}\right)\enspace {\delta }_{n,{m}_{1}+{m}_{2}}\end{equation*}$$

and taking the generating function of both sides of equation (A3) with respect to n ⩾ 1 one gets B_⩾0(s) − 1 = B_⩾0(s)(B_>0(s) − 1) which yields

$$\begin{equation}{B}_{{\geqslant}0}\left(s\right)=\frac{1}{2-{B}_{{ >}0}\left(s\right)}.\end{equation} \tag{ A4 }$$

Now, we consider a bridge of length n ⩾ 1. Let n₁ and n₂ respectively denote the first and last time at which it reaches its minimum before arriving at x_n = 0. Clearly 0 ⩽ n₁ ⩽ n₂ ⩽ n − 1. For n₁ = 0, the bridge is a positive excursion of probability b_⩾0(n) which we write like in equation (A3) with m = n₂. For 1 ⩽ n₁ ⩽ n − 1, we do the Vervaat construction [45] by cutting at n₁. One obtains

$$\begin{align}\hfill \mathrm{Prob.}\left({x}_{n}=0\right)& =\sum _{{n}_{1}=0}^{n-1}\sum _{{n}_{2}={n}_{1}}^{n-1}{b}_{{\geqslant}0}\left({n}_{2}-{n}_{1}\right){b}_{{ >}0}\left(n-{n}_{2}+{n}_{1}\right)\hfill \\ \hfill & =\sum _{{n}_{1}=0}^{n-1}\sum _{\ell =0}^{n-{n}_{1}-1}{b}_{{\geqslant}0}\left(\ell \right){b}_{{ >}0}\left(n-\ell \right),\hfill \end{align} \tag{ A5 }$$

where ℓ = n₂ − n₁. Now, write ${\sum }_{\ell =0}^{n-{n}_{1}-1}={\sum }_{\ell =0}^{n}-{\sum }_{\ell =n-{n}_{1}}^{n}$ on the right-hand side of equation (A5), use the relation

$$\begin{equation*}\sum _{\ell =0}^{n}{b}_{{\geqslant}0}\left(\ell \right){b}_{{ >}0}\left(n-\ell \right)=2{b}_{{\geqslant}0}\left(n\right),\end{equation*}$$

(which follows trivially from equation (A3)) in the sum ${\sum }_{\ell =0}^{n}$, and make the change of variable ℓ → n − ℓ in the sum ${\sum }_{\ell =n-{n}_{1}}^{n}$. One gets

$$\begin{align}\hfill \mathrm{Prob.}\left({x}_{n}=0\right)& =\sum _{{n}_{1}=0}^{n-1}\left(2{b}_{{\geqslant}0}\left(n\right)-\sum _{\ell =0}^{{n}_{1}}{b}_{{\geqslant}0}\left(n-\ell \right){b}_{{ >}0}\left(\ell \right)\right)\hfill \\ \hfill & =2n\enspace {b}_{{\geqslant}0}\left(n\right)-\sum _{\ell =0}^{n-1}\left(n-\ell \right)\enspace {b}_{{\geqslant}0}\left(n-\ell \right){b}_{{ >}0}\left(\ell \right)\hfill \\ \hfill & =2n\enspace {b}_{{\geqslant}0}\left(n\right)-\sum _{{\ell }_{1}{\geqslant}1}\sum _{{\ell }_{2}{\geqslant}0}{\ell }_{1}{b}_{{\geqslant}0}\left({\ell }_{1}\right){b}_{{ >}0}\left({\ell }_{2}\right)\enspace {\delta }_{n,{\ell }_{1}+{\ell }_{2}}.\hfill \end{align} \tag{ A6 }$$

The factor (n − ℓ) in the sum over ℓ in the second line comes from the fact that the double sum over ℓ and n₁ is over the domain 0 ⩽ ℓ ⩽ n₁ ⩽ n − 1: for each value of ℓ, n₁ takes (n − ℓ) values. Taking the generating function of both sides of equation (A6) with respect to n ⩾ 1, one finds

$$\begin{equation}\sum _{n{\geqslant}1}\mathrm{Prob.}\left({x}_{n}=0\right)\enspace {t}^{n}=t\enspace {B}_{{\geqslant}0}^{\prime }\left(t\right)\left(2-{B}_{{ >}0}\left(t\right)\right)=\frac{t\enspace {B}_{{\geqslant}0}^{\prime }\left(t\right)}{{B}_{{\geqslant}0}\left(t\right)},\end{equation} \tag{ A7 }$$

where we have used equation (A4). It remains to divide both sides of equation (A7) by t and integrate over t from t = 0 to t = s, with B_⩾0(0) = 1. One obtains

$$\begin{equation}{B}_{{\geqslant}0}\left(s\right)=\mathrm{exp}\left(\sum _{n{\geqslant}1}\mathrm{Prob.}\left({x}_{n}=0\right)\frac{{s}^{n}}{n}\right).\end{equation} \tag{ A8 }$$

Finally, from $Z\left(s\right)=\sqrt{{B}_{{\geqslant}0}\left(s\right)}$, the relation $\mathrm{Prob.}\left({x}_{n}=0\right)=\left(\alpha /2\pi \right){\int }_{-\pi /\alpha }^{\pi /\alpha }\hat{f}{\left(k\right)}^{n}\mathrm{d}k$, and $\hat{f}\left(k\right)=\hat{f}\left(-k\right)$, one obtains

$$\begin{equation}\begin{aligned}\hfill Z\left(s\right)& =\mathrm{exp}\left(\frac{1}{2}\sum _{n{\geqslant}1}\mathrm{Prob.}\left({x}_{n}=0\right)\frac{{s}^{n}}{n}\right)\hfill \\ \hfill & =\mathrm{exp}\left(-\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1-s\hat{f}\left(k\right)\right]\mathrm{d}k\right).\hfill \end{aligned}\end{equation} \tag{ A9 }$$

In this appendix, we derive the scaling form given in equation (82) for any discrete jump distribution with Lévy index 0 < μ ⩽ 2. Write P(h = 0) = p, P(h = 1) = p₁, and u_m = P(h = m)/p₁. From $\hat{f}\left(k\right)=p+2{p}_{1}{\sum }_{m{\geqslant}1}{u}_{m}\enspace \mathrm{cos}\left(m\alpha k\right)$ and p + 2p₁∑_m⩾1u_m = 1, one gets

$$\begin{equation}\hat{f}\left(k\right)=1-\frac{\left(1-p\right)}{{\sum }_{\mathfrak{m}{\geqslant}1}{u}_{m}}\enspace \sum _{m{\geqslant}1}{u}_{m}\left[1-\mathrm{cos}\left(m\alpha k\right)\right].\end{equation} \tag{ B1 }$$

Introducing ${\hat{f}}_{0}\left(k\right)$ the Fourier transform of the jump distribution corresponding to p = 0 with fixed u_m (m ⩾ 1), we rewrite equation (B1) in the simpler form

$$\begin{equation}\hat{f}\left(k\right)=1-\left(1-p\right)\left[1-{\hat{f}}_{0}\left(k\right)\right].\end{equation} \tag{ B2 }$$

Using the expression (B2) of $\hat{f}\left(k\right)$ on the right-hand side of equation (57) and considering the scaling regime s, p → 1 with fixed (1 − p)/(1 − s), one obtains

$$\begin{equation}Z\left(s\right)\sim \frac{{Z}_{0}\left(1\right)}{\sqrt{1-p}}\enspace \mathrm{exp}\left(-\frac{\alpha }{2\pi }\enspace {\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1+\frac{\left(1-s\right)/\left(1-p\right)}{1-{\hat{f}}_{0}\left(k\right)}\right]\mathrm{d}k\right),\end{equation} \tag{ B3 }$$

where Z₀(1) is given by equation (43) with s = 1 and ${\hat{f}}_{0}\left(k\right)$ instead of $\hat{f}\left(k\right)$. Now, inverting the generating function in equation (50), one gets the integral representation

$$\begin{equation}E\left({R}_{n}\right)=\frac{1}{2i\pi }\enspace \oint \frac{\mathrm{d}s}{{s}^{n+1}{\left(1-s\right)}^{3/2}Z\left(s\right)}.\end{equation} \tag{ B4 }$$

Making the change of variable s = exp(−λ/n) in equation (B4), using the fact that, in the n → +∞ limit, only the vicinity of s = 1 contributes to the s-integral, and taking for Z(s) the asymptotic expression in equation (B3), one obtains the following scaling form for E(R_n),

$$\begin{equation}E\left({R}_{n}\right)\sim \mathcal{G}\left[\left(1-p\right)\enspace n\right],\end{equation} \tag{ B5 }$$

valid for n → +∞, p → 1, and fixed (1 − p)n = O(1), with the scaling function

$$\begin{equation}\mathcal{G}\left(x\right)=\frac{\sqrt{x}}{2i\pi \enspace {Z}_{0}\left(1\right)}\enspace {\int }_{\mathcal{L}}\mathrm{exp}\left(\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1+\frac{\lambda /x}{1-{\hat{f}}_{0}\left(k\right)}\right]\mathrm{d}k\right)\enspace \frac{{\mathrm{e}}^{\lambda }}{{\lambda }^{3/2}}\mathrm{d}\lambda .\end{equation} \tag{ B6 }$$

The large argument behavior of $\mathcal{G}\left(x\right)$ is obtained by doing the same analysis as the one for the behavior of equation (57) near s = 1. Making the substitutions (1 − s) → λ/x and $\hat{F}\left(k\right)\to 1/\left(1-{\hat{f}}_{0}\left(k\right)\right)$ in the integral on the right-hand side of equation (57), and using the small k behavior ${\hat{f}}_{0}\left(k\right)\sim 1-{\left({a}_{0}k\right)}^{\mu }$, with a₀ = a/(1 − p)^1/μ independent of p, one finds

$$\begin{equation}\begin{aligned}\hfill \mathcal{G}\left(x\right)& \sim \frac{\sqrt{x}}{2i\pi \enspace {Z}_{0}\left(1\right)}\enspace {\int }_{\mathcal{L}}\left(1+\frac{\alpha }{2{a}_{0}}\sqrt{\frac{\lambda }{x}}\enspace {\delta }_{\mu ,\enspace 2}\right)\enspace \frac{{\mathrm{e}}^{\lambda }}{{\lambda }^{3/2}}\mathrm{d}\lambda \hfill \\ \hfill & =\frac{2}{{Z}_{0}\left(1\right)}\sqrt{\frac{x}{\pi }}+\frac{\alpha }{2{a}_{0}{Z}_{0}\left(1\right)}\enspace {\delta }_{\mu ,\enspace 2}\quad \left(x\to +\infty \right),\hfill \end{aligned}\end{equation} \tag{ B7 }$$

where we have kept the terms surviving the x → +∞ limit, only. Injecting this result onto the right-hand side of equation (B5) and using the relations ${Z}_{0}\left(1\right)=Z\left(1\right)\sqrt{1-p}$ and a₀Z₀(1) = aZ(1), one can check that the scaling form (B5) coincides with the large n behavior (65) in the limit (1 − p)n ≫ 1.

On the other hand, the small argument behavior of $\mathcal{G}\left(x\right)$ is obtained from the equation (B6) as

$$\begin{align}\hfill \mathcal{G}\left(x\right)& \sim \frac{\sqrt{x}}{2i\pi \enspace {Z}_{0}\left(1\right)}\enspace {\int }_{\mathcal{L}}\mathrm{exp}\left(\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[\frac{\lambda /x}{1-{\hat{f}}_{0}\left(k\right)}\right]\mathrm{d}k\right)\enspace \frac{{\mathrm{e}}^{\lambda }}{{\lambda }^{3/2}}\mathrm{d}\lambda \hfill \\ \hfill & =\frac{1}{{Z}_{0}\left(1\right)}\enspace \mathrm{exp}\left(-\frac{\alpha }{2\pi }{\int }_{0}^{\pi /\alpha }\mathrm{ln}\left[1-{\hat{f}}_{0}\left(k\right)\right]\mathrm{d}k\right)=1\quad \left(x\to 0\right).\hfill \end{align} \tag{ B8 }$$

Using this result on the right-hand side of equation (B5), one finds E(R_n) ∼ 1 in the opposite limit (1 − p)n ≪ 1. All these results hold for any discrete jump distribution with Lévy index 0 < μ ⩽ 2.