Limits of renewal processes and Pitman-Yor distribution

We consider a renewal process with regularly varying stationary and weakly dependent steps, and prove that the steps made before a given time $t$, satisfy an interesting invariance principle. Namely, together with the age of the renewal process at time $t$, they converge after scaling to the Pitman--Yor distribution. We further discuss how our results extend the classical Dynkin--Lamperti theorem.


Introduction
By one of the main results in renewal theory, it is known that the age of a renewal process has a limiting distribution, given that its steps have finite mean. When the steps are iid and regularly varying with infinite mean, the limiting distribution is determined by the Dynkin-Lamperti theorem. In this article, we aim to understand the limiting behavior of the whole path of such a renewal process before a given time t. Moreover, we do that under milder conditions, that is, we keep the regular variation assumption, but allow certain degree of dependence between the steps of the renewal process. More precisely, we assume that the steps form a stationary sequence (Y n ) of nonnegative random variables which are regularly varying with index α ∈ (0, 1). By one characterization of regular variation, see Resnick [16], this means that there exists a sequence of nonnegative real numbers (d n ) such that as n → ∞, where v −→ denotes vague convergence of measures on (0, ∞) and the limiting measure satisfies µ(x, ∞) = x −α for all x > 0.
It will be useful in the sequel to extend (d n ) to a function on [0, ∞) by denoting d(t) = d ⌊t⌋ , for t ≥ 0, with d 0 = 1. It is known that d has an asymptotic inverse, d say, see Seneta [18], in the sense that d( d(t)) ∼ d(d(t)) ∼ t , (1.2) as t → ∞. One can show that d is a regularly varying function with index α. Denote by the first passage time of the level t by the random walk with steps (Y n ). Our main goal is to describe the asymptotics of all the steps in the renewal process before the passage time τ (t), i.e. of the random variables including the age of the renewal process at the passage time, that is (1.5) For iid steps (Y n ), the proof of the following classical theorem can be found in Bingham et al. [5].
as t → ∞, where the random variable on the right hand side has a generalized arcsine distribution with the density It turns out that the necessity part of this theorem holds for certain dependent renewal processes too. More importantly, in all such cases one can describe the joint asymptotic behavior of the random variables in (1.4) and (1.5), and show that they, when ordered, form a sequence which converges towards the so-called Pitman-Yor distribution. As far as we know, this result is new even in the iid case.
The paper is organized as follows: in Section 2 we consider Pitman-Yor distribution on the interval partitions from the perspective of point processes theory. We further present two limiting theorems about stationary strongly mixing sequences (Y n ) satisfying (1.1) which are likely to be of independent interest. These theorems are used in Section 3 to determine the asymptotic distribution of the steps Y i /t , i = 1, . . . , τ (t) − 1 and the age of the renewal process A (t) . We also exhibit how this result extends the classical Dynkin-Lamperti theorem and discuss corresponding assumptions. It immediately yields the joint asymptotic distribution for the ranked lengths of excursions in a simple symmetric random walk, cf. Csáki and Hu [6]. More technical proofs and results concerning Skorohod's topology and convergence of point measures are postponed to the Appendix.

Point processes and Pitman-Yor distribution
In a remarkable series of papers: [14], [13], [12]; Perman, Pitman and Yor describe the distribution of jumps of stable subordinators on a given time interval. In particular, Pitman and Yor in [14], use such jumps to introduce a new family of distributions on interval partitions and relate them to the classical arcsine laws for Brownian motion. Recall that a stable subordinator (S(t)) t≥0 is a Lévy process with the Laplace transform given by the formula for some α ∈ (0, 1) and a constant c α > 0 which turns out to be unimportant in the sequel. So without loss of generality we typically assume c α = 1, i.e. µ ′ = µ. The subordinator (S(t)) has the distribution of the inverse local times of d-dimensional Bessel process, for d = 2(1 − α), with the case α = 1/2 corresponding to the Brownian motion. In other words, jumps of the process (S(t)) correspond to the lengths of excursions of the Brownian motion or, more generally Bessel process, away form the origin. By Ito's representation (S(t)) can be constructed from a Poisson process N on the space [0, ∞) × (0, ∞] with intensity measure equal to Leb × µ ′ , so that We alternatively say that N is a Poisson random measure and denote this by N ∼ PRM(Leb × µ ′ ). By the construction, (S(t)) is a nondecreasing element of the space of càdlàg functions D[0, ∞). Denote by z ← the right continuous generalized inverse of a function z ∈ D[0, ∞), i.e.
The generalized inverse of the process S(t) is the process It is well defined and continuous at any s ≥ 0, and for the reasons explained above it is called local time process by Bertoin in [3]. If we denote by Z the closure of the range of the process (S(t)), the maximal open subintervals in the set Z c ∩ (0, s), s > 0, correspond to the jumps of the subordinator before it crosses over level s. Their lengths are (S(t) − S(t−)) , t < L(s), which are equal to P i , T i < L(s), above, together with the last incomplete jump which has the length 2) see Bertoin [3]. Considering these points in descending order we arrive at the sequence Observe that the distribution of V (s) corresponds to the distribution of the point process Clearly, normalizing the infinite sequence V (s) by s > 0 produces a random sequence which sums up to one. An extraordinary observation of Pitman and Yor [14] was that This is surprising, since the sequence on the right hand side is produced by ordering and scaling the points and therefore has no special "last interval" as in (2.2). Due to the identity (2.3), it suffices to describe the distribution of V (1), thus we denote The distribution of this sequence corresponds to the distribution of the point process It turns out to be easier to describe the distribution of the size-biased permutation of the sequence V (1), say Perman [11] proved that for a sequence of independent random variables ξ i = 1, 2, 3, . . ., such that ξ ∼ Beta(1 − α, iα). We call the distribution of the sequence V (1), or equivalently of the point process M (α) , the Pitman-Yor distribution with parameter α. This distribution has further natural extension to two parameter family of distributions on the interval partitions, see Pitman and Yor [15]. That family found important applications in nonparametric Bayesian statistics, e.g. see Teh and Jordan [19] and references therein. Moreover, the arcsine laws for the fraction of time Brownian motion spends in the upper halfplane at a fixed time t or at inverse local time L(s), can be seen as corollaries of the results in [14].
In the course of showing (2.3), Pitman and Yor showed that U 1 above actually has the distribution of the final interval length A 1 from (2.2). This distribution is the same as the generalized arcsine distribution of the random variable A in theorem 1.1. These results allowed Perman [12] to describe the density of sup{P i : T i < L(1)}, which corresponds to the longest excursion of the d-dimensional Bessel process completed by the time 1, and of D 1 = max {sup{P i : T i < L(1)}, A 1 }, which has the same interpretation but includes the last a.s. incomplete excursion.
It is well known that iid sequence (Y n ) satisfies (1.1), if and only if the following convergence of point processes holds where N denotes a Poisson process on the space [0, ∞) × (0, ∞] with intensity measure Leb × µ, see Resnick [16]. The convergence of point processes in (2.4) and throughout is to be understood with respect to the vague topology on the If (1.1) and (2.4) hold for a general stationary sequence (Y n ), then it necessarily has the extremal index equal to 1, see Leadbetter et al. [9] for instance. In other words, the partial maxima in the sequence (Y n ) behave as if the sequence was iid, i.e. M n = max{Y 1 , . . . , Y n } satisfies M n /d n → Φ α , as n → ∞ where Φ α denotes the standard Fréchet distribution, i.e. Φ α (x) = exp(−x −α ), x > 0. Next theorem, proved in the Appendix, claims that the opposite is also true. Namely, strongly mixing sequences which satisfy (1.1) and have extremal index equal to 1, necessarily satisfy (2.4). Observe that the theorem holds for all α > 0, and not merely on the interval (0, 1) which is of our main interest in this paper.
Theorem 2.1. Suppose that (Y n ) is a stationary strongly mixing sequence of nonnegative regularly varying random variables with tail index α > 0. Then For iid steps, (2.4) and the continuous mapping theorem imply in D[0, ∞) with respect to Skorohod's J 1 metric, see Resnick [17], Chapter 7, cf. theorem 2.2 below. Moreover, for α ∈ (0, 1) has finite value with probability 1 for all t > 0. Observe that Denote In the following theorem we show that this convergence is joint with the convergence in (2.5), whenever (2.4) holds.
Theorem 2.2. Suppose that (Y n ) is a stationary strongly mixing sequence of nonnegative regularly varying random variables with extremal index equal to one and the tail index α ∈ (0, 1). Then, as t → ∞ By the continuous mapping argument, see Lemma 4.1 in the Appendix, it follows that The random variable L in (2.6) represents the first passage time of the level one by the α-stable subordinator S. Its distribution is known in the literature as a Mittag-Leffler distribution.

Main theorem
Our main result extends the sufficiency part of the Dynkin-Lamperti theorem in a couple of ways. We first show that one can describe the limiting distribution of not merely the age of the renewal process at time t, but also the behavior of all other large steps before that time. By doing that, we obtain the Pitman-Yor distribution as the limiting distribution for the steps after appropriate normalization. We also show that the statement of Dynkin-Lamperti theorem about iid regularly varying random variables can be generalized to cover all regularly varying sequences with non-clustering extremes considered in the previous section. For simplicity, denote Theorem 3.1. Suppose that (Y n ) is a stationary strongly mixing sequence of nonnegative regularly varying random variables with extremal index equal to one and the tail index α ∈ (0, 1). Then, as t → ∞,

1)
where M (α) represents a Pitman-Yor point process with parameter α. Moreover, the convergence above is joint with as t → ∞, where A has the generalized arcsine distribution given in (1.6).
For a measure ν on a measurable space (S, S), by ν| B we denote the restriction of the measure ν on the set B ∈ S given by ν| B (C) = ν(B ∩ C), C ∈ S. Abusing this notation somewhat, for any time period . (3.2) Proof. By theorem 2.2, (N t , S t ) d −→ (N, S) in the appropriate product topology, as t → ∞. Moreover, the limit (N, S) a.s. satisfies the regularity assumption of lemma 4.1 and theorem 4.1 below. Therefore, this convergence is joint with the convergence in By theorem 4.1, as t → ∞, In particular for f ∈ C + K ((0, ∞]), where C + K denotes the family of nonnegative continuous functions with compact support as t → ∞. Since the corresponding Laplace functionals converge, we conclude that Because, this holds jointly with Remark 3.1. The strong mixing assumption in the theorem is actually unnecessarily strong, one could alternatively consider any stationary sequence (Y n ) which satisfies (2.4). In the extreme value theory it is known that this holds under milder conditions cf. Basrak et al. [1].
Remark 3.2. An interesting implication of theorem 3.1 concerns the lengths of excursions of the simple symmetric random walk during the first n steps. They are known to be independent and regularly varying with index α = 1/2. Therefore, theorem can be applied to deduce and extend results in Csáki and Hu [6] about the asymptotic distribution of these excursions.
The value A (t) in theorem 3.1 is called the undershoot or the age of the renewal process at time t. Similarly, one could define the overshoot at t as Recall that L (t) represents the scaled first passage time. Straightforward application of theorem 2.2 and lemma 4.1 yields the following corollary which should be compared with Dynkin-Lamperti theorem, cf. theorem 8.6.3 of Bingham et al. [5]. Note however that the corollary admits weak dependence between the steps of the renewal process.

Appendix
Proof. (of theorem 2.1) As we explained before the theorem, it remains to show sufficiency. Assume that (Y n ) is strongly mixing with the extremal index equal to 1. Denote by α(n) the mixing coefficients of the sequence (Y n ). Then set l n = ⌊max{1, n 0.1 }⌋, clearly l n = o(n), l n → ∞. Introduce also the sequence r n = ⌊max{1, n α(l n ), n 2/3 }⌋, and observe r n → ∞. By proposition 1.34 in Krizmanić [8], the sequence (r n ) satisfies the following condition: for every f ∈ (4.1) as n → ∞, assuming without loss of generality that the support of f lies in [0, L] × [l, ∞] for L, l > 0. In other words, the strong mixing condition implies the condition A ′ (a n ) introduced in Basrak et al. [1].
Observe that lim n→∞ (P (Y ≤ d n u)) n = lim for any u > 0. Note that the sequences (l n ) and (r n ) satisfy r n = o(n) nα(l n ) = o(r n ), l n = o(r n ). According to O'Brien [10], the extremal index θ of the sequence (Y n ) satisfies for any fixed u > 0. Since, by assumption, θ = 1, we obtain as n → ∞. Hence, by stationarity, for every u > 0, Consequently (Y n ) is jointly regularly varying in the sense of Basrak and Segers [2]. Moreover, its tail sequence is trivial.
We observe next that by (4.1) and (4.3), the point processes for all measurable sets I, J. Vague convergence theory as presented in section 3.4 of Resnick [16], shows that this mapping is continuous. This turns to be useful, since Observe that as t → ∞. Hence, by (4.6) and (1.2), one can conclude Assume that these values are finite for each u > 0, but tend to ∞ as u → ∞. This makes s t and s well defined, unbounded, nondecreasing elements of the space of càdlàg functions D[0, ∞). Their right continuous generalized inverses (or hitting time functions) we denote by s ← and s ← t , recall that s ← (u) = inf{v ∈ [0, ∞) : s(v) > u} , u ≥ 0 . We will use the following abbreviations in the sequel It is well known that J 1 convergence in general does not imply convergence of s t towards s at a given point. However, the following technical lemma shows that under some regularity conditions, it implies the convergence of both s t and its left limit at the first passage time τ t . It is a consequence of Theorem 13.6.4 in Whitt [20] which has a weaker assumption that s t converge towards s in M 2 topology. and suppose that s(v) < 1 for each v < τ . Then  Proof. By the definition of the vague convergence, it is enough to show the convergence in the state space [0, ∞) × (u, ∞] for some number u > 0 in the arbitrary neighborhood of 0. Because n is a Radon measure, one can always find u > 0 which is arbitrarily close to 0 and satisfies n([0, ∞) × {u}) = 0 and u < s(τ ) − s(τ −). Since s is a càdlàg function, there exists ε 0 > 0, such that Since s t and s are monotone functions, by theorem 2.15 a) in Jacod and Shiryayev [7], chapter VI, (4.9) implies that there exists a dense set of points G ⊆ [0, ∞) such that s t (d) → s(d) for each d ∈ G. Take 0 < ε ≤ ε 0 , such that τ + ε ∈ G, n({τ + ε} × (0, ∞)) = 0, and such that (4.12) By Proposition 3.13 in Resnick [16], there exists a constant t ′ 0 , such that for all
By the assumption, there exists ε > 0 such that s(τ ) > 1 + ε. Note that there also exists d ∈ G, d < τ such that For all sufficiently large t now σ t > d, and