Universality of local times of killed and reflected random walks

In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Knight(1963) who has shown that for the symmetric simple random walk local times converge weakly towards a squared Bessel process. Our result can be seen as an extension of the second Ray-Knight theorem to all asymptotically stable random walks.


Introduction
Let {S n } be a random walk on Z with increments {X k } which are independent copies of a random variable X. Let τ − be the first weak descending ladder epoch of our random walk, that is, τ − := min{n ≥ 1 : S n ≤ 0}.
We shall always assume that EX = 0. This implies that {S n } is recurrent and, in particular, τ − is almost sure finite. Let A := {1 < α < 2; |β| ≤ 1} ∪ {α = 2, β = 0} be a subset in R 2 . For (α, β) ∈ A and a random variable X write X ∈ D (α, β) if the distribution of X belongs to the domain of attraction of a stable law with characteristic function This means that there exists an increasing, regularly varying with index 1/α function c(x) such that S n /c(n) converges in distribution towards the stable law given by (1). By c −1 (x) we shall denote the inverse to c(x) function. Clearly, c −1 is regularly varying wih index α. Let L(n, x) = n j=0 I(S j = x), x ∈ Z denote the local time of the process {S n }. We first consider local times of {S n } killed at leaving positive half-axis.
In order to formulate our result we have to introduce some notation. Let τ + be the first strict ascending ladder epoch and let χ ± denote ladder heights corresponding to τ ± . Define where {χ ± j } are independent copies of χ ± . Finally, let h ± denote the mass functions of H ± , that is, Theorem 1. If X ∈ D(α, β), then there exists c α,β such that, as N → ∞, for every fixed x ≥ 0, and where The exponential distribution in (2) is not surprising. Indeed, if {S n } hits N before it leaves (0, ∞) then we may assume that {S n } starts at N . Thus the conditioned distribution of the local time is independent of the starting point. Let p N denote the probability that S n becomes negative before it returns to N . Clearly, p N is positive. Then Consequently, (2) is equivalent to It is immediate from the definition of U (x, N ) that But for positive start points x one needs additional restrictions.
The assumption that h + is long-tailed is not easy to check. Furthermore, (4) can used only if we know the asymptotic behaviour of h + (N ). In other words, we need a strong renewal theorem for positive ladder heights. Some sufficient conditions for this theorem can be found in [2] and [12].
For random walks with finite second moments h + is asymptotically constant and, moreover, one can provide an exact expression for the constant c α,β . and We now turn to reflected random walks. More precisely, we shall look at local times of the process W n+1 = (W n + X n+1 ) + , n ≥ 0 which starts at zero, that is, W 0 = 0. Set T 0 = 0 and define recursively T n+1 := min{k > T n : W k = 0}, n ≥ 0.
We are interested in the asymptotic behaviour of local times Let M (N ) be a sequence of natural numbers. For every N define a rescaled process Theorem 4. Assume that X ∈ D(α, β). If h + is regularly varying then there exists a process L α,β = {L α,β (u), u ≥ 0} such that, for any sequence Remark 5. It follows easily from the proof of Theorem 4 that the marginals of L α,β are compound Poisson distributions with exponentially distributed jumps. ⋄ The distribution of the limiting process is known for α = 2 only. Knight [10] has shown that if S n is a simple random walk then l (N ) converges weakly (and not only in the sence of finite dimensional distributions) towards the square of 0-dimensional Bessel process. Since the limit is the same for all random walks belonging to the domain of attraction of the normal distribution, we conclude that L 2,0 is the squared 0-dimensional Bessel process. More precisely, L 2,0 is the unique strong solution of the equation where {B(t), t ≥ 0} is the Brownian motion. Unfortunately, we are not able to determine the limit for α < 2. Using Corollary 3.5 from Eisenbaum and Kaspi [3] one can give a characterisation of L α,β in terms of permanental processes. If, additionally, the limiting stable process is symmetric, that is, β = 1 then L α,β can be described by a squared Gaussian process.
The proof in [10] is based on a trick which works for simple random walks only. Let X n be Rademacher random variables, that is, P(X n = ±1) = 1/2. Fix some m ≥ 1 and for every n > 0 let Q (m) n denote the number times k < T m+1 such that W k−1 = n and W k = n + 1. Set also Q (m) 0 = m. The key observation in [10] is that {Q (m) n , n ≥ 0} is a Markov chain with transition kernel given by Now it is immediate that Q (m) n is a martingale. The markovian structure and the martingale property allowed Knight to prove that converges weakly towards the squared Bessel process. Noting that n−1 , we then conclude that the limit for l (N ) is equal to the limit for q (N ) multiplied by 2. In other words, the limit for l (N ) is again a squared Bessel process, but with a different scaling constant c.
Rogers [11] noticed that (7) corresponds to a critical Galton-Watson process Z n with the geometric offspring distribution and Z 0 = m. Then one has also L W (T m+1 , n) = Z n + Z n−1 . As a consequence, convergence of local times follows from the corresponding results for branching processes. The idea of connecting local times and branching processes has been recently used by Hong and Yang [9], who have extended Knight's result to all left-continuous random walks with bounded jumps. This has been achieved via connecting local times to a 2-type critical branching process. Similar to Knight's paper, this embedding into a branching process ensures markovian and martingale properties, which help to prove weak convergence.
Our approach is based on the derivation of asymptotics for mixed moments of local times, which seems to go back at least to Darling and Kac [4]. But in order to apply this method to killed random walks one needs to know asymptotic behaviour of the corresponding Green (renewal) function. It has become possible due to the recent result by Caravenna and Chaumont [1] on bridges of random walks conditioned to stay positive. This method allows to prove convergence of finite dimensional distributions in a strightforward manner. But at the moment we do not know how to prove the tightness of the sequence l (N ) . The was not a problem in papers [10,9], where the martingal structure of Galton-Watson processes can be used.

Asymptotic behaviour of moments of local times
The following renewal theorem is crucial for our proof.
Proposition 6. Assume that x N /N → u > 0 and y N /N → v > 0. Then, there exists a α,β (u, v) > 0 such that Proof. We split the sum in (8) into three parts In view of the Gnedenko local limit theorem, P x (S n = z) ≤ C c n , for any x, z ∈ Z and n ≥ 1. Therefore, Further, applying Corollary 13 from Doney [6], we get

This yields
Further, applying (4.2) from Caravenna and Chaumont [1] to every summand in the second sum, we obtain where Y t is a stable process with Y 1 defined by (1). Combining this with (9) and (10) and letting ε → 0, we obtain Thus, the proof is complete.
Remark 7. The limit a α,β is the so-called 0-potential density of a killed stable process. ⋄ We next give an explicit expression for a 2,0 . Using the reflection principle for the Brownian motion, one can easily obtain Therefore, Substituting x = y −1 we obtain According to formula 3.434(1) from [8] ∞ Consequently, We now turn to higher moments of local times.
and lim N →∞ Proof. It is immediate from the Markov property that Applying Proposition 6, we get (13). The proof of (14) is identical.
Remark 10. The right hand side of (16) is a specialisation of Kac's moment formula (see [4]) for local times of a killed stable process.

Proof. It is clear that
Similarly, Taking expectations and applying (13) and (14) we get the desired result.
It remains to consider the case when the random walk starts at a fixed point x. Replacing u 0 N by x in the first equality of (15), we get Thus, additionally to Proposition 6, we have to determine the asymptotic behaviour of ∞ j1=1 P x (S j1 = u 1 N, τ − > j 1 ). First, by the duality lemma for random walks, P 0 (S j = l, τ − > j) = P(S j = l, j is a strict ascending ladder epoch).
Consequently, for each l ≥ 1, Second, for every positive x we split τ − into descending ladder epochs. Then, using the Markov property and (18), we get Combining this with (17) and Corollary (6), one can easily obtain Proposition 11. As N → ∞, 3. Proof of Theorem 1.
Recall that the distribution of L(τ − , N ) conditioned on L(τ − , N ) > 0 does not depend on the starting point. Using Proposition 9 with u 0 = u 1 = . . . = u m = 1, we conclude that Thus, by the method of moments, Since for all k ≥ 1 and all x, we get (2). As we have shown in the previous section, Furthermore, by the Markov property, Consequently, .

Proof of Theorem 2
We start with the Laplace transform of the vector Obviously, for every r ≥ 1, From this we infer that and Let {µ = (µ 1 , µ 2 , . . . , µ m )} be the set of m-dimensional multiindices. Then, by the binomial formula, Recall that h + is assumed to be regularly varying. Thus, it follows from Proposition 11 that there exists ψ j (u, µ) such that Furthermore, this proposition gives also the following bound for φ j Combining (22) and (23), we obtain with some function ψ j satisfying This estimate is immediate from (22) respectively. Note that estimate (26) allows to let r → ∞ for all λ k small enough. As a result, there exists δ > 0 such that if λ k ∈ [0, δ) for all k then lim N →∞ where Ψ(u, λ) := ∞ j=1 (−1) j j! ψ j (u, λ).
Notice also that (22)  for any sequence M (N ) whichis asymptotically equivalent to nh + (N )/c − (N ). By the continuity theorem for the Laplace transformation, the distribution of the vector (l (N ) (u 1 ), l (N ) (u 2 ), . . . , l (N ) (u m )) converges weakly to a law F u , which is characterised by the Laplace transform λ → e −Ψ(u,λ) . The continuity of functions Ψ(u, λ) implies, in particular, the consistency of the family of finite dimensional distributions {F u }. Thus, the proof is completed.