Functional Central Limit Theorems for Occupancies and Missing Mass Process in Infinite Urn Models

We study the infinite urn scheme when the balls are sequentially distributed over an infinite number of urns labeled 1,2,... so that the urn j at every draw gets a ball with probability pj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p_j$$\end{document}, where ∑jpj=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum _j p_j=1$$\end{document}. We prove functional central limit theorems for discrete time and the Poissonized version for the urn occupancies process, for the odd occupancy and for the missing mass processes extending the known non-functional central limit theorems.


Introduction
In this paper, we study the following classical urn model first considered by Karlin [12]: n ≥ 1 balls are distributed one by one over an infinite number of urns enumerated from 1 to infinity. The ball distributed at step j = 1, 2 . . . , call it jth ball, gets into urn i with probability p i , ∞ i=1 p i = 1, independently of the other balls. Such multinomial occupancy schemes arise in many different applications, in Biology [11], Computer science [13,14] and in many other areas, see, e.g., [10] and the references therein.
Let X j be the urn the jth ball gets into and let J i (n) be the number of balls the ith urn contains after n balls are distributed: Of a particular interest is the asymptotic behavior of the following quantities: the number of urns containing at least k ≥ 1 balls and containing exactly k balls: the number of urns with an odd number of balls and the scaled missing mass introduced in [12]: We also use notation R n def = R * n,1 = k≥1 R n,k for the number of non-empty urns. Renumbering the urns if necessary, we may assume that the sequence ( p i ) i≥1 is monotonely decaying. We further assume that it is regularly varying: where L(x) is a slowly varying function as x → ∞. Following Karlin's [12] original approach, we will consider a Poissonized version of the model when the balls are put into urns at the times of jumps of a homogeneous Poisson point processes (s), s ≥ 0 with intensity 1 on R + . According to the independent marking theorem for Poisson processes, {J i ( (s)) def = i (s), s ≥ 0} are independent homogeneous Poisson processes with intensities p i . To ease the notation, we write simply and we introduce the following Poissonized version of the scaled missing mass: It differs from M (s) by the scaling factor s vs. (s), but, when properly scaled, it is asymptotically equivalent to it. Ordinary (not functional) central limit theorems for the above quantities were established under various conditions in [2,3,9,10,[12][13][14]. In particular, under rather general conditions on the sequence ( p i ) involving an unbounded growth of the variances, the following results are available: a strong law of large numbers and asymptotic normality of R n , an asymptotic normality of the vector (R n,1 , . . . , R n,ν ), local limit theorems, etc.
We acknowledge a novel method of a randomized decomposition for proving FCLTs developed in a recent paper [8], but we do not use it here. As a particular case of their Theorem 2.3, a FCLT holds for the processes R n and U n when θ ∈ (0, 1).
Our goal here is to establish a FCLT for the triplet of processes: the occupancy, odd occupancy and the scaled missing mass when θ ∈ (0, 1]. In particular, we obtain previously unknown FCLT for U n for θ = 1 and for M n when θ ∈ (0, 1]. Up to a normalizing constant, the FCLT stated in Theorem 1 also holds for the original (nonscaled) missing mass ∞ i=1 p i 1I J i (n)=0 on any interval t ∈ [ε, 1], ε > 0, separated from 0. The paper extends the results of [6] and [7], where a functional central limit theorem (FCLT) was shown under condition (3) for the vector process 1] in the case θ ∈ (0, 1]. Extending the FCLT to the case θ = 0 would require additional to (3) conditions. As it was mentioned in [12] and in [2], θ = 0 does not imply that the variances grow to infinity and various asymptotic behavior is possible for different statistics. We also argue that even an infinite growth of variances does not guarantee per se the required relative compactness.
When θ = 1, we need a function It is known (see [12]) that L * (x) is slowly varying when x → ∞.
Finally, for t ∈ [0, 1] introduce the following notation: We are now ready to formulate the main result of the paper.

Proof of Theorem 1
We start with formulating a couple of lemmas proved in [7]. We will generally use the letter C and its variants to denote a constant whose value is of no importance for us and note in parentheses the parameters it depends upon. This should not lead to a confusion when the same notation is used for, actually, different constants in different contexts, the same way O(1) notation is used.
Lemma 2 For any ε, δ ∈ (0, 1) there exists an N = N (ε, δ) such that for any n ≥ N , In preparation of the proof, let us introduce some further notation and establish a few inequalities we will be using.
In view of (5), let For any two positive τ 1 ≤ τ 2 , define and their expectations are denoted by Similarly for M, Clearly, for all natural k, Similarly, As a result, We are using the same notation u i , m i and u i , m i without explicitly specifying the corresponding values of τ 1 < τ 2 ; this should not create a confusion. The following lemma will be used in the proof of the relative compactness of the process M * n (t).
Proof Put τ 2 = nt 2 and τ 1 = nt 1 . Since the variance of an indicator does not exceed its expectation, we have that 3 .
Step 1: Covariance The first rather technical step consists in establishing a formula for the covariances which is put in Appendix.
Step 2: Convergence of finite-dimensional distributions Along the lines of the proof of [9,Th. 12], one can show that for the triangular array of m-dimensional vectors (i.e., independent in k for every n) satisfies the Lindeberg condition (see, e.g., [5,Th. 6.2]). Similarly, the convergence of the finite-dimensional distributions is shown for the process M * n (t).

Step 3: Relative compactness
We shall follow the following plan: (a) prove the continuity of the limiting process; (b) prove that U * n and U * * n (M * n and M * * n ) are sufficiently close; (c) prove the relative compactness of U * * n (M * * n ). a(U) Take τ 1 = nt 1 , τ 2 = nt 2 for 0 < t 1 < t 2 < 0. Then, We have used above the independence of the summands, inequality (8) and Lemma 1.
Since the covariance function has a limit, [1, Th. 1.4] will imply that the limiting Gaussian process a.s. has a continuous modification on [0, 1]. Since the trajectories of the limiting Gaussian process belong a.s. to the class C(0, 1), the weak convergence in the Skorohod topology implies the weak convergence in the uniform metric, see, e.g., [4]. Therefore, it is sufficient to prove the relative compactness of {U * n } n≥n 0 (with n 0 as in Lemma 1) in the Skorohod topology. b(U) Since with probability one we have Hence, for all η > 0,
Above, we have used (11) in the first inequality, (8) in the second and finally (10) and Lemma 1 alongside with the bound If t 2 − t 1 ≥ 1/n, then there are the following three cases: the same inequality yields we have that  C(0, 1), the weak convergence in the Skorohod topology implies the uniform convergence, see [4]. Thus, it is sufficient to prove a relative compactness of the family {M * n } n≥n 0 in the Skorohod topology (here, n 0 is the same as in Lemma 1).

b(M)
Set τ 2 = nt and τ 1 = [nt]. Since τ 2 − τ 1 ≤ 1, Let m i = m i (τ 1 , τ 1 + 1) and m i = E m i . Then, we have almost surely We know that for any integer k ≥ 2 Using the independence of the terms and Rosenthal inequality, for any k ≥ 2, Hence, for k ≥ [2/θ ] + 1 and all η > 0 Therefore, it is sufficient to show the local compactness of {M * * n } n≥n 0 in the Skorohod topology.
Above, we have used inequalities (9), (10) and Lemmas 3, 1 alongside with the bound When t 2 − t 1 ≥ 1/n, we have the following three cases: , then the Cauchy-Schwarz inequality gives , is similar to the previous case. Thus, the required compactness follows from [4,Th. 13.5].
Finally, for the next step we need to show that M(s), when time scaled, is close to its fully Poissonized version Namely, we aim to show that where as s → ∞ and it is bounded by 1. Thus, there exists a sufficiently small ε = ε(θ) > 0 such that for δ n = n ε−1 when n → ∞.
Step 4: Approximation of the initial process Since (t) is monotone, the strong law of large numbers implies that for any ε, δ ∈ (0, 1) there is an integer N = N (ε, δ) such that for all n ≥ N one has see Lemma 2. Here and below, F stands for R, U or M. The relative compactness of the distributions {F * n } n≥n 0 implies that for any ε ∈ (0, 1) and η > 0 there exist δ ∈ (0, 1) and an integer N 1 = N 1 (ε, η) such that for all n ≥ N 1 , Hence, since which proves Theorem 1.