A Limit Theorem For a Nested Inﬁnite Occupancy Scheme in Random Environment

We investigate an inﬁnite balls-in-boxes scheme, in which boxes are arranged in nested hierarchy and random probabilities of boxes are deﬁned in terms of iterated fragmentation of a unit mass. Gnedin and Iksanov (2020) obtained a multivariate functional central limit theorem with centering for the cumulative occupancy counts as the number of balls becomes large. We prove a counterpart of their result, in which centering is not needed and the limit processes are not Gaussian. An application is given to the scheme generated by a residual allocation model


Introduction
The infinite occupancy scheme is an urn model that has numerous applications to statistics, combinatorics, and computer science. It is often depicted as a balls-in-boxes model. One throws balls successively and independently into an infinite array of boxes 1, 2, . . ., so that each ball hits box k with positive probability p k , and k∈N p k = 1. This classical model is usually called Karlin's occupancy scheme because of Karlin's seminal contribution (Karlin 1967). Features of the occupancy pattern emerging after the first n balls are thrown have been intensively studied, see Gnedin, Hansen, and Pitman (2007) and Iksanov (2016) for surveys.
There is also a randomized version of the classical infinite occupancy scheme, in which the hitting probabilities of boxes are positive random variables (P k ) k∈N with an arbitrary joint distribution satisfying k∈N P k = 1 almost surely (a.s.). We consider here a variant of this occupancy scheme, which corresponds to a nested family of boxes. The construction is conveniently described in terms of the genealogical structure of populations. Let I 0 := {∅} be the initial ancestor and I 1 := {1, 2, . . .} be the set of the first generation boxes with some random hitting probabilities P 1 , P 2 , . . .. Divide now each box i into subboxes i1, i2, . . . and define the hitting probabilities of the subboxes by where (P (i) k ) k∈N is an independent copy of (P k ) k∈N . These subboxes are interpreted as the second generation boxes which form the set I 2 . We repeat this procedure for boxes of each generation until an ∞-ary tree of nested boxes ∪ k∈N 0 I k has been constructed. Here, N 0 = N ∪ {0}. Note that the hitting probabilities of each generation boxes sum up to one by construction, that is, v∈I j P (v) = 1 a.s. for each j ∈ N.
We assume that the random probabilities of boxes and the outcome of throwing balls are defined on a common probability space. For n, j, r ∈ N, denote by K n,j,r the number of the jth generation boxes υ ∈ I j containing exactly r out of n first balls, and let be a cumulative count of occupied boxes, where · is the ceiling function. With probability one, the random function u → K n,j (u) is non-decreasing and right-continuous, hence belongs to D[0, 1]. Here and hereafter, for an interval I ⊆ [0, ∞) we denote by D(I) the Skorokhod space on I, that is, the set of all càdlàg functions defined on I.

Related literature
The nested occupancy scheme in random environment is introduced in Bertoin (2008). Among other things, it is proved there that, with both n and j diverging to infinity in such a way that j ∼ a log n, where a is some positive constant, K n,j (1) the number of occupied boxes in the jth generation, properly normalized, satisfies a central limit theorem with random centering as the number n of balls thrown tends to ∞. Let H n,j denote the index of the first generation in which all the boxes have less then j balls, when n balls have been thrown and G n,j denote the index of the first generation in which there exists a box containing less then j balls. In Joseph (2011) the a.s. asymptotic behaviour of these two random variables is investigated when n → ∞. It turns out that H n,j and G n,j are of the order log n as n → ∞ and their asymptotics depend on j, when j is smaller than the critical value j * , and do not depend on j, when j is larger than j * . Businger (2017) is concerned with similar problems in a more general setting (the underlying branching process is a multitype weighted branching process rather than a weighted branching process). The three aforementioned articles investigated late generations of the occupancy scheme in random environment, that is, generations whose indices j are of the order log n. Iksanov and Mallein (2022) provide a full classification of regimes of the a.s. convergence for the number of occupied boxes in the late generations, thereby complementing the earlier results obtained in Bertoin (2008).
In Gnedin and Iksanov (2020) early generations (whose indices j do not depend on n) of the nested occupancy scheme in random environment are treated. In particular, a multivariate functional central limit theorem for the process ((K n,1 (u)) u∈[0, 1] , (K n,2 (u)) u∈[0, 1] , . . .), properly scaled, centred and normalized, is proved, with the scaling limit of each coordinate being an a.s. continuous Gaussian process. Under the assumption that the hitting probabilities of boxes are given by a residual allocation model (a.k.a. stick-breaking) weak convergence of finite-dimensional distributions of (K n, jnt (1)) t>0 , properly normalized and centered, is obtained in Iksanov, Marynych, and Samoilenko (2022d) for j n → ∞ and j n = o((log n) 1/2 ) as n → ∞. A weaker version with j n = o((log n) 1/3 ) as n → ∞ appeared earlier in Buraczewski, Dovgay, and Iksanov (2020).
Observe that the sequence ((− log P (v)) v∈I j ) j∈N forms a branching random walk. In general, the definition of a branching random walk does not require that v∈I 1 P (v) = 1 a.s. A branching random walk with (− log P (v)) v∈I 1 being a globally perturbed random walk (see Section 6 for the definition) or a standard random walk is called an iterated perturbed random walk or an iterated standard random walk. Closely related to the present setting are the articles Bohun, Iksanov, Marynych, and Rashytov (2022), Iksanov, Rashytov, and Samoilenko (2023), Iksanov, Marynych, and Rashytov (2022c) and Iksanov, Kabluchko, and Kotelnikova (2022b), in which various aspects of iterated perturbed random walks and iterated standard random walks are investigated.
Multivariate functional central limit theorems for the number of boxes containing at least 1 ball, 2 balls, . . . and the number of boxes containing 1 ball, 2 balls, . . ., properly scaled, normalized and centered, are proved in Iksanov, Kabluchko, and Kotelnikova (2022a) and Iksanov and Kotelnikova (2022), respectively for a nested occupancy scheme with deterministic probabilities.

Main result
Our purpose is to prove a functional limit theorem for ((K n,1 (u)) u∈[0, 1] , (K n,2 (u)) u∈[0, 1] , . . .), properly scaled and normalized without centering. One consequence of our assumptions is that the scaling limit of each component is an a.s. nondecreasing process, which particularly cannot be Gaussian. Thus, even though our setting and that of Gnedin and Iksanov (2020) look similar, the ideas exploited here and in Gnedin and Iksanov (2020) are quite different at places.

For the given fragmentation law
Here is a basic recursive decomposition, which will be essentially used throughout the paper: j−1 (t)) t≥0 , . . . are independent copies of (N j−1 (t)) t≥0 , which are also independent of T 1 , T 2 , . . .. Passing in (1) to the expectations yields which shows that V j is the j-fold Lebesgue-Stieltjes convolution of V with itself.
Throughout the paper we write ⇒ for weak convergence in a function space. Now we formulate the assumptions of our main result: for some α ≥ 0 and some slowly varying at ∞; and in the J 1 -topology on D[0, ∞), where (W (u)) u≥0 is an a.s. locally Hölder continuous process with exponent β ∈ (0, 1]. The latter means that, for every T > 0 there exists an a.s. finite random variable M T such that, for all x, y ∈ [0, T ], where the integral exists as a pathwise Lebesgue-Stieltjes integral. Observe that W 1 = W .
We are ready to state our main result.
The remainder of the paper is structured as follows. Section 4 collects a number of preparatory results, which are then used in Section 5 for the proof of Theorem 1. An application of Theorem 1 to the occupancy scheme in random environment generated by a residual allocation model is given in Section 6.

Auxiliary results
In this section we collect several auxiliary results.

Lemma 3. Suppose
Proof. We use mathematical induction. According to (4) inequality (9) holds with j = 1. Assuming its validity for j = i − 1 we shall show that (9) also holds with j = i.
We first provide a convenient representation for Using (1) we obtain The last equality is a consequence of Here, while the expression in the first parantheses is G s -measurable, the conditional expectation of the expression in the second parantheses is equal to 0 a.s., for T s is G s -measurable and N Further, having utilized monotonicity of V i−1 for the last inequality and Lemma 2 for the asymptotic relation. Using monotonicity of y → E(N i−1 (y)) 2 and Lemma 2 we infer as t → ∞. This completes the proof of (10).
Let (Y k ) k∈N 0 be a sequence of nonnegative random variables. Put M (t) = k≥0 1 {Y k ≤t} for t ≥ 0 and assume that M (t) < ∞ a.s. for t ≥ 0. For a function h ∈ D[0, ∞), put In the terminology of the article Iksanov and Rashytov (2021) (X(t)) t≥0 is a general shot noise process with the response function h and the counting process M .
We state a slightly corrected version of Theorem 1 in Iksanov and Rashytov (2021). The formulation in Iksanov and Rashytov (2021) contains an extra assumption, which is omitted here.

Proof of the main result
Theorem 1 is an immediate consequence of the following two results.

Proposition 6. Suppose
where P → denotes convergence in probability.
Proof of Theorem 5. We claim it suffices to prove weak convergence of finite-dimensional distributions. According to the Cramér-Wold device, the task amounts to showing that, for any i, m ∈ N, any nonnegative β r, k , r, k ∈ N, r ≤ i, k ≤ m and any nonnegative u r, k , r, k ∈ N, where d → denotes one-dimensional distributional convergence. Since the product J 1 -topology on (D[0, ∞)) N is used, tightness of the distributions of the converging processes is secured by tightness of the distributions of their coordinates. The latter is ensured by Remark 2.1 in Yamazato (2009), because, for each j ∈ N and each t > 0, the random function u → N j (ut) is a.s. nondecreasing on [0, ∞) and the limit process W j is a.s. continuous (this can be easily checked).
Using (1), write, for t ∈ R and j ≥ 2, We shall prove that and, for each j ∈ N, These limit relations entail (12). Note that, in view of the regular variation of V j , (14) guarantees that, for each u ≥ 0, Proof of (13). Since Let (t n ) n∈N be an arbitrary sequence of positive numbers satisfying lim n→∞ t n = ∞. According to the Skorokhod representation theorem there exist processes ((Ñ tn (y)) y≥0 ) n∈N having the same distribution as ((N (t n y)/V (t n )) y≥0 ) n∈N and a processW having the same distribution as W , such that lim n→∞Ñ tn (y) =W (y) a.s. on D[0, ∞).
The distribution of the left-(right-) hand side of (15) does not change upon replacing N (t n y)/V (t n ) withÑ tn (y) (W withW ). Hence, (15) follows if we can show that According to (5), the first sum on the left-hand side converges a.s. to the first sum on the right-hand side. For r ≥ 2, the process Y r is an instance of a general shot noise process with the response function V r−1 and the counting process N , see the paragraph preceding Proposition 4 for the definition. The function V r−1 is nondecreasing and, by Lemma 2, regularly varying at ∞ of index α(r − 1). The function 1/V is nonincreasing and, by assumption, is regularly varying at ∞ of index −α. The process W is an a.s. locally Hölder continuous with exponent β ∈ (0, 1]. Thus, the process Y r satisfies all the assumptions of Proposition 4, and an application of this result yields by Lemma 2. This entails that each term on the left-hand side of (16) which corresponds to a single r = 2, 3, ..., i converges a.s. to the corresponding term on the right-hand side of (16). This proves (16) and thereupon (13).
It follows from (9) that Var (N j−1 (t)) ≤ E(N j−1 (t)) 2 ≤ C(V j−1 (t)) 2 for some positive constant C and t ≥ 1. Hence, Here, the second inequality is justified by monotonicity of V j−1 and (2), and the last equality is secured by Lemma 2.
Proof of Proposition 6. Fix any j ∈ N. By Proposition 3.6 in Gnedin and Iksanov (2020), for n ∈ N, where y 0 ∈ (0, 1) is a deterministic constant which does not depend on n, nor on (P (v)) v∈I j .
In view of Theorem 5 and Lemma 2, To complete the proof, according to Markov's inequality, it suffices to show that the expectation of each of the first three terms in (17) divided by V j (log n) converges to 0 as n → ∞.
For the first term, this follows from the fact, which is a consequence of Lemma 2, that the function t → V j (log t) is slowly varying at ∞. For the second term, this is trivial. For the third term, write having utilized slow variation of t → V j (log t) for the last equality. The penultimate equality is justified by Lebesgue's dominated convergence theorem in combination with Potter's bound (Theorem 1.5.6(i) in Bingham, Goldie, and Teugels (1987)): for all x ≥ 1 and large n, V j (log(nx))/V j (log n) ≤ 2x 1/2 .
Let (ξ k , η k ) k∈N be independent copies of a random vector (ξ, η) with positive arbitrary dependent components. Denote by (S k ) k∈N 0 the zero-delayed standard random walk with increments ξ k , that is, S 0 := 0 and S k := ξ 1 + . . . + ξ k for k ∈ N. Put The random sequence (T * k ) k∈N is known in the literature as a (globally) perturbed random walk, see Section 1 in Iksanov (2016). With P k as in (18), that is, (T k ) k∈N is a particular instance of the perturbed random walk with (ξ, η) = (| log U |, | log(1 − U )|). (20) Throughout this section, N (t) = k≥1 1 {T k ≤t} and V (t) = EN (t) = k≥1 P{T k ≤ t}, t ∈ R for the particular (T k ) k∈N as above.
As a consequence, We shall apply this limit relation to (ξ, η) as in (20).