From infinite urn schemes to decompositions of self-similar Gaussian processes

We investigate a special case of infinite urn schemes first considered by Karlin (1967), especially its occupancy and odd-occupancy processes. We first propose a natural randomization of these two processes and their decompositions. We then establish functional central limit theorems, showing that each randomized process and its components converge jointly to a decomposition of certain self-similar Gaussian process. In particular, the randomized occupancy process and its components converge jointly to the decomposition of a time-changed Brownian motion $\mathbb B(t^\alpha), \alpha\in(0,1)$, and the randomized odd-occupancy process and its components converge jointly to a decomposition of fractional Brownian motion with Hurst index $H\in(0,1/2)$. The decomposition in the latter case is a special case of the decompositions of bi-fractional Brownian motions recently investigated by Lei and Nualart (2009). The randomized odd-occupancy process can also be viewed as correlated random walks, and in particular as a complement to the model recently introduced by Hammond and Sheffield (2013) as discrete analogues of fractional Brownian motions.


Introduction
We consider the classical infinite urn schemes, sometimes referred to as the balls-in-boxes scheme. Namely, given a fixed infinite number of boxes, each time a label of the box is independently sampled according to certain probability µ, and a ball is thrown into the corresponding box. This model has a very long history, dating back to at least Bahadur [1]. For a recent survey from the probabilistic point of view, see Gnedin et al. [11]. In particular, the sampling of the boxes forms naturally an exchangeable random partition of N. Exchangeable random partitions have been extensively studied in the literature, and have connections to various areas in probability theory and related fields. See the nice monograph by Pitman [22] for random partitions and more generally combinatorial stochastic processes. For various applications of the infinite urn schemes in biology, ecology, computational linguistics, among others, see for example Bunge and Fitzpatrick [6].
In this paper, we are interested in a specific infinite urn scheme. More precisely, we consider µ as a probability measure on N := {1, 2, . . .} which is regularly varying with index 1/α, α ∈ (0, 1). See the definition in Section 2.1. This model was first considered by Karlin [14] and we will refer to it by the Karlin model in the rest of the paper.
We start by recalling the main results of Karlin [14]. Let (Y i ) i≥1 represents the independent sampling from µ for each round i ≥ 1, and Y n,k := n i=1 1 {Y i =k} be the total counts of sampling of the label k in the first n rounds, or equivalently how many balls thrown into the box k in the first n rounds. In particular, Karlin investigated the asymptotics of two statistics: the total number of boxes that have been chosen in the first n rounds, denoted by Z * (n) := The processes Z * and U * are referred to as the occupancy process and the odd-occupancy process, respectively. While Z * is a natural statistics to consider in view of sampling different species, the investigation of U * is motivated via the following light-bulb-switching point of view from Spitzer [26]. Each box k may represent the status (on/off) of a light bulb, and each time when k is sampled, the status of the corresponding light bulb is switched either from on to off or from off to on. Assume that all the light bulbs are off at the beginning. In this way, U * (n) represents the total number of light bulbs that are on at time n. Central limit theorems have been established for both processes in [14], in form of (1) Z * (n) − EZ * (n) σ n ⇒ N (0, σ 2 Z ) and U * (n) − EU * (n) σ n ⇒ N (0, σ 2 U ) for some normalization σ n , with σ 2 Z and σ 2 U explicitly given as the variances of the limiting normal distributions, and where ⇒ denotes convergence in distribution. We remark that σ 2 n is of the order n α , up to a slowly varying function at infinity. The next seemingly obvious task is to establish the functional central limit theorems for the two statistics. However, to the best of our knowledge, this has not been addressed in the literature. Here by functional central limit theorems, or weak convergence, we are thinking of results in the form of (in terms of Z * ) in space D([0, 1]) for some normalization sequence σ n and a Gaussian process Z * . In view of (1) and the fact that σ 2 n has the same order as n α , the scaling limit Z * , if exists, is necessarily self-similar with index α/2.
In this paper, instead of addressing this question directly, we consider a more general framework by introducing the randomization to the Karlin model (see Section 2.1 for the exact definitions). The randomization of the Karlin model reveals certain rich structure of the model. In particular, it has a natural decomposition. Take the randomized occupancy process Z ε for example. We will write Z ε (n) = Z ε 1 (n) + Z ε 2 (n) and prove a joint weak convergence result in form of In other words, the limit trivariate Gaussian process (Z 1 (t), Z 2 (t), Z(t)) t∈[0,1] can be constructed by first considering two independent Gaussian processes Z 1 and Z 2 with covariance to be specified, and then set Z(t) := Z 1 (t)+Z 2 (t), t ∈ [0, 1]; in this way its finite-dimensional distributions are also determined. We refer such results as weak convergence to the decomposition of a Gaussian process. Similar results for the randomized odd-occupancy process are also obtained. Here is a brief summary of the main results of the paper. • As expected, various self-similar Gaussian processes appear in the limit. In this way, the randomized Karlin model and its components, including Z * and U * as special quenched cases, provide discrete counterparts of several self-similar Gaussian processes. These processes include notably the fractional Brownian motion with Hurst index H = α/2, the bi-fractional Brownian motion with parameter H = 1/2, K = α, and a new self-similar process Z 1 . • Moreover, in view of the weak convergence to the decomposition, the randomized Karlin model are discrete counterparts of certain decompositions of self-similar Gaussian processes. The randomized occupancy process and its two components converge weakly to a new decomposition of the time-changed Brownian motion (B(t α )) t≥0 , α ∈ (0, 1) (Theorem 1). The randomized odd-occupancy process and its two components converge weakly to a decomposition of the fractional Brownian motion with Hurst index H = α/2 ∈ (0, 1/2) (Theorem 2). This decomposition is a particular case of the decompositions of bi-fractional Brownian motion recently discovered by Lei and Nualart [17]. Self-similar processes have been extensively studied in probability theory and related fields [9], often related to the notion of long-range dependence [25,21]. Among the selfsimilar processes arising in the limit in this paper, the most widely studied one is the fractional Brownian motion. Fractional Brownian motions, as generalizations of Brownian motions, have been widely studied and used in various areas of probability theory and applications. These processes are the only centered Gaussian processes that are self-similar with stationary increments. The investigation of fractional Brownian motions dates back to Kolmogorov [16] and Mandelbrot and Van Ness [18]. As for limit theorems, there are already several models that converge to fractional Brownian motions in the literature. See Davydov [7], Taqqu [27], Enriquez [10], Klüppelberg and Kühn [15], Peligrad and Sethuraman [20], Mikosch and Samorodnitsky [19], Hammond and Sheffield [12] for a few representative examples. A more detailed and extensive survey of various models can be found in Pipiras and Taqqu [21]. Besides, we also obtain limit theorems for bi-fractional Brownian motions introduced by Houdré and Villa [13]. They often show up in decompositions of self-similar Gaussian processes; see for example [24,17]. However, we do not find other discrete models in the literature. As for limit theorems illustrating decompositions of Gaussian processes as ours do, in the literature we found very few examples; see Remark 4. Our results connect the Karlin model, a discrete-time stochastic process, to several continuous-time self-similar Gaussian processes and their decompositions. By introducing new discrete counterparts, we hope to improve our understanding of these Gaussian processes. In particular, the proposed randomized Karlin model can also be viewed as correlated random walks, in a sense complementing the recent model introduced by Hammond and Sheffield [12] that scales to fractional Brownian motions with Hurst index H ∈ (1/2, 1). Here, the randomized odd-occupancy process (U ε below) is defined in a similar manner, and scales to fractional Brownian motions with H ∈ (0, 1/2).
The paper is organized as follows. Section 2 introduces the model in details and present the main results. Section 3 introduces and investigates the Poissonized models. The de-Poissonization is established in Section 4.

Randomization of Karlin model and main results
2.1. Karlin model and its randomization. We have introduced the original Karlin model in Section 1. Here, we specify the regular variation assumption. Recall the definition of (p k ) k≥1 . We assume that p k is non-increasing, and define the infinite counting measure ν on [0, ∞) by where max ∅ = 0. Following Karlin [14], the main assumption is that ν(t) is a regularly varying function at ∞ with index α in (0, 1), that is for all x > 0, lim t→∞ ν(tx)/ν(t) = x α , or equivalently where L(t) is a slowly varying function as t → ∞, i.e. for all x > 0, lim t→∞ L(tx)/L(t) = 1.
For the sake of simplicity, one can think of p k ∼ k→∞ Ck − 1 α for some α ∈ (0, 1) and a normalizing constant C > 0.
We have introduced two random processes considered in Karlin [14]: the occupancy process and the odd-occupancy process as Z * (n) := k≥1 1 {Y n,k =0} and U * (n) := k≥1 1 {Y n,k is odd} respectively. To introduce the randomization, let ε := (ε k ) k≥1 be a sequence of i.i.d. Rademacher random variables (i.e. P(ε k = 1) = P(ε k = −1) = 1/2) defined on the same probability space as the (Y n ) n≥1 and independent from them. In the sequel, we just say that ε is a Rademacher sequence in this situation (and thus implicitly, ε will always be independent of (Y n ) n≥1 ).
Let ε be a Rademacher sequence. We introduce the randomized occupancy process and the randomized odd-occupancy process by We actually will work with decompositions of these two processes given by with for all k ≥ 1 and n ≥ 1, In the preceding definitions, the exponent ε refers to the randomness given by the Rademacher sequence (ε k ) k≥1 . Nevertheless, in some of the following statements, the sequence of (ε k ) k≥1 can be chosen fixed (deterministic) in {−1, 1} N . Then the corresponding processes can be considered as "quenched" versions of the randomized process. For this purpose, it is natural to introduce the centering with p k (n) and q k (n) respectively above. Actually, we will establish quenched weak convergence for Z ε 1 and U ε 1 (see Theorem 3 and Remark 1). With a little abuse of language, for both cases we keep ε in the notation and add an explanation like 'for a Rademacher sequence ε' or 'for all fixed ε ∈ {−1, 1} N ', respectively.

Main results.
As mentioned in the introduction, we are interested in the scaling limits of the previously defined processes. We denote by D([0, 1]) the Skorohod space of cadlag functions on [0, 1] with the Skorohod topology (see [2]). Throughout, we write σ n := n α/2 L(n) 1/2 , where α and L are the same as in the regular variation assumption (4). Observe that ν(n) = L(n) = σ n = 0 for n < 1/p 1 . Therefore, when writing 1/σ n we always assume implicitly n ≥ 1/p 1 . We obtain similar results for Z ε and U ε . Below are the main results of this paper.
Z 1 and Z 2 are independent, and they have covariances U 1 and U 2 are independent, and they have covariances To achieve these results, we will first prove the convergence of the first (Z ε 1 and U ε 1 ) and the second (Z ε 2 and U ε 2 ) components, respectively. For the first components we have the following stronger result.
where Z 1 and U 1 are as in Theorems 1 and 2.
Remark 1. Theorem 3 is a quenched functional central limit theorem. In particular, when taking ε = 1 = (1, 1, . . . ) ∈ N, Theorem 3 recovers and generalizes the central limit theorems for Z * (n) and U * (n) established in Karlin [14] (formally stated in (1)): the (nonrandomized) occupancy and odd-occupancy processes of the Karlin model scale to the continuous-time processes Z 1 and U 1 , respectively. Moreover, as the limits in Theorem 3 do not depend on the value of ε, this implies the annealed functional central limit theorems (the same statement of Theorem 3 remains true for a Rademacher sequence ε), and entails essentially the joint convergence to the decomposition. Now we take a closer look at the processes appearing in Theorem 1 and Theorem 2 and the corresponding decompositions. The decomposition of U is a special case of the general decompositions established in Lei and Nualart [17] for bi-fractional Brownian motions. Recall that a bi-fractional Brownian motion with parameter H ∈ (0, 1), K ∈ (0, 1] is a centered Gaussian process with covariance function It is noticed in [17] that one can write where the left-hand side above is a multiple of the covariance function of a fractional Brownian motion with Hurst index HK, and the second term on the right-hand side above is positive-definite and hence a covariance function. Therefore, (8) induces a decomposition of a fractional Brownian motion with Hurst index HK into a bi-fractional Brownian motion and another self-similar Gaussian process. Comparing this to Theorem 2, we notice that our decomposition of U corresponds to the special case of (8) with H = 1/2, K = α. Up to a multiplicative constant, U is a fractional Brownian motion with Hurst index H = α/2. The process U 1 is the bi-fractional Brownian motion with H = 1/2, K = α, and it is also known as the odd-part of the two-sided fractional Brownian motion; see Dzhaparidze and van Zanten [8]. That is where B α/2 is a two-sided fractional Brownian motion on R with Hurst index α/2 ∈ (0, 1). The process U 2 admits a representation where (B(t)) t∈[0,1] is the standard Brownian motion. It is shown that U 2 (t) has a version with infinitely differentiable path for t ∈ (0, ∞) and absolutely continuous path for t ∈ [0, ∞). At the same time, U 2 also appears in the decomposition of sub-fractional Brownian motions [4,24]. For the decomposition of Z in Theorem 1, to the best of our knowledge it is new in the literature. Remark that Z is simply a time-changed Brownian motion (Z(t)) t≥0 The latter is not surprising as the coefficients q k (n) and p k (n) have the same asymptotic behavior. However, we cannot find related reference for Z 1 in the literature. The following remark on Z 1 has its own interest. Remark 2. The process Z 1 may be related to bi-fractional Brownian motions as follows. One can write where Z 1 and Z 2 are as before and independent, and V is a centered Gaussian process with covariance . Therefore, as another consequence of our results, we have shown that for the bi-fractional Brownian motions, the covariance function R H,K in (7) is well defined for H = 1/(2α), K = α for all α ∈ (0, 1). The range α ∈ (0, 1/2] is new.
To prove the convergence of each individual process, we apply the Poissonization technique. Each of the Poissonized processesZ ε 1 ,Z ε 2 ,Ũ ε 1 ,Ũ ε 2 is an infinite sum of independent random variables, of which the covariances are easy to calculate, and thus the finitedimensional convergence follows immediately. The hard part for the Poissonized processes is to establish the tightness forZ ε 1 andŨ ǫ 1 . For this purpose we apply a chaining argument. Once the weak convergence for the Poissonized models are established, we couple the Poissonized models with the original ones and bound the difference. The second technical challenges lie in this de-Poissonization step. Remark that Karlin [14] also applied the Poissonization technique in his proofs. Since he only worked with central limit theorems and us the functional central limit theorems, our proofs are more involved.
Remark 3. One may prove the weak convergence ( 1] directly, without using the decomposition. We do not present the proofs here as they do not provide insights on the decompositions of the limiting processes.
Remark 4. We are not aware of other limit theorems for the decomposition of processes in a similar manner as ours, but with two exceptions. One is the symmetrization well investigated in the literature of empirical processes [28]. Take for a simple example the empirical distribution function with uniform (0, 1) distribution. By symmetrization one considers an independent Rademacher sequence ε and It is straight-forward to establish √ n (F ε n (t), F ε,1 n (t), F ε,2 n (t)) t∈ The other example of limit theorems for decompositions is the recent paper by Bojdecki and Talarczyk [5] who provided a particle-system point of view for the decomposition of fractional Brownian motions. The model considered there is very different from ours, and so is the decomposition in the limit.
2.3. Correlated random walks. We first focus our discussion on U ε . One can interpret the process U ε as a correlated random walk by writing (9) U ε (n) = X 1 + · · · + X n , for some random variables (X i ) i≥1 taking values in {−1, 1} with equal probabilities, and the dependence among steps is determined by a random partition of N. Viewing X i as the step of a random walk at time i, U ε in (9) then represents a correlated random walk. To have such a representation, recall (Y i ) i≥1 and consider the random partition of N induced by the equivalence relation that i ∼ j if and only if Y i = Y j ; that is, the integer i and j are in the same component of the partition if and only if the i-th and j-th balls fall in the same box. Once (Y n ) n≥1 is given and thus all components are determined, one can define (X n ) n≥1 as follows. For each k ≥ 1, suppose all elements in component k (defined as {i : Y i = k}) are listed in increasing order n 1 < n 2 < · · · , and set X n 1 := ε k and iteratively X n ℓ+1 := −X n ℓ . In this way, it is easy to see that each X n is taking values in {−1, 1} with equal probabilities, and conditioning on (Y n ) n≥1 , X i and X j are completely dependent if i ∼ j, and independent otherwise. The verification of (9) is straight-forward. The above discussion describes how to construct a correlated random walk from random partitions in two steps. The first is to sample from a random partition. The second is to assign ±1 values to (X n ) n≥1 conditioned on the random partition sampled. A similar interpretation can be applied to another model of correlated random walks introduced in Hammond and Sheffield [12]. The Hammond-Sheffield model also constructed a collection of random variables taking values in {−1, 1}, of which the dependence is determined by a random partition of Z, in form of a random forest with infinitely many components indexed by Z. There are two differences between the Hammond-Sheffield model and the randomized odd-occupancy process U ε : first, the underlying random partition is different: notably, the random partition in the infinite urn scheme is exchangeable, while this is not the case for the random partition introduced in Z in [12]; rather, the random partition there inherits certain long-range dependence property, which essentially determines that the Hurst index in the limit must be in (1/2, 1). Second, for X i in the same component of random partitions, Hammond-Sheffield model set them to take the same value (all 1 or all −1 with equal probabilities), independently on each component.
The alternative way of assigning values for X i in the same component is the key idea in our framework. Clearly this has been considered by Spitzer [26] and Karlin [14], if not earlier. Actually, Hammond and Sheffield [12] suggested, as an open problem, to apply the alternative way of assigning values to their model and asked whether the modified model scales to fractional Brownian motions with Hurst index in (0, 1/2). In our point of view, in order to obtain a discrete model in the similar flavor of the Hammond-Sheffield model that scales to a fractional Brownian motion with Hurst index H ∈ (0, 1/2), the alternative way of assigning values is crucial, while the underlying random forest with long memory is not that essential. Our results support this point of view. Actually, looking for a model in a similar spirit to complement the Hammond-Sheffield model as the discrete counterparts of fractional Brownian motions was another motivation for this paper. At the same time, the aforementioned suggestion in [12] remains a challenging model to analyze.
As for the occupancy process, similarly one can view Z ε as a correlated random walk. The random partition being the same, this time for each component k with elements n 1 < n 2 < · · · , we set X n 1 = ǫ k , X n i = 0, i ≥ 2 to obtain Z ε = X 1 + · · · + X n .
The dependence of this random walk is simpler than the odd-occupancy process.

Poissonization
Recall that we are interested in the processes Z ε and U ε , and instead to deal with them directly we work with the decompositions Z ε = Z ε 1 + Z ε 2 and U ε = U ε 1 + U ε 2 with the components defined in (5) and (6). 3.1. Definitions and preliminary results. The first step in the proofs is to consider the Poissonized versions of all the preceding processes in order to deal with sums of independent variables. Let N be a Poisson process with intensity 1, independent of the sequence (Y n ) n≥1 and of the Rademacher sequence ε considered before. We set Then the processes N k , k ≥ 1, are independent Poisson processes with respective intensity p k . We now consider the Poissonized processes, for all t ≥ 0, These Poissonized randomized occupancy and odd-occupancy processes have similar decompositions as the original processes Using the independence and the stationarity of the increments of Poisson processes, we derive the following useful identities. For all 0 ≤ s ≤ t and all k ≥ 1, Note that, in particular, the functionsp k andq k are sub-additive. Further, we will have to deal with the asymptotics of the sums over k of thep k orq k . For this purpose, recall that (see [14,Theorem 1]) the assumption (4) implies We will need further estimates on the asymptotics of V (t) as stated in the following lemma.

Now we introduce
Let δ > 0 be such that α + δ < 1 and α − δ > γ. Observe that L * has the same asymptotic behavior as L by definition. In addition, L * is bounded away from 0 and ∞ on any compact set of [0, ∞). Thus, by Potter's theorem (see [3, Theorem 1.5.6]) there exists a constant C δ > 0 such that for all x, y > 0 We infer, uniformly in t ∈ [0, 1], and both integrals are finite (the second one because we have taken δ such that α + δ < 1). Further, t α−δ ≤ t γ for all t ∈ [0, 1] and thus the lemma is proved.
3.2. Functional central limit theorems. We now establish the invariance principles for the Poissonized processes.
where Z 1 is as in Theorem 1 and U 1 is as in Theorem 2.
Proof. In the sequel ε ∈ {−1, 1} N is fixed. The proof is divided into three steps.
(i) The covariances. Using the independence of the N k , and that ε 2 k = 1 for all k ≥ 1, we infer that for all 0 ≤ s ≤ t, whence by (12), For the odd-occupancy process, using the independence and the stationarity of the increments of the Poisson processes, for 0 ≤ s ≤ t, Thus, again by (12), (ii) Finite-dimensional convergence. The finite-dimensional convergence for both processes is a consequence of the Lindeberg central limit theorem, using the Cramér-Wold device. Indeed, for any choice of constants a 1 , . . . , a d ∈ R, d ≥ 1, and any reals t 1 , . . . , t d ∈ [0, 1], the independent random variables ε k , k ≥ 1, n ≥ 1 are uniformly bounded. This entails the finite-dimensional convergence for (Z ε 1 (nt)/σ n ) t∈[0,1] . The proof for (Ũ ε 1 (nt)/σ n ) t∈[0,1] is similar. (iii) Tightness. The proof of the tightness is technical and delayed to Section 3.3.

Proposition 2. For any Rademacher sequence
where Z 2 is as in Theorem 1 and U 2 is as in Theorem 2.
(i) The covariances. Since the ε k are independent, using (12), we have for all t, s ≥ 0 (ii) Finite-dimensional convergence. Since Z ε 2 is a sum of independent bounded random variables, the finite-dimensional convergence follows from the Cramér-Wold device and the Lindeberg central limit theorem.

Tightness forZ
To show the tightness, we will prove The tightness then follows from the Corollary of Theorem 13.4 in [2]. To prove (13), we first show the following two lemmas.
A chaining argument is then applied to establish the tightness by proving the following. Proof of Lemma 2. We prove for G =Ũ ε 1 . The case G =Z ε 1 can be treated in a similar way and is omitted. In view of Lemma 1 it is sufficient to prove that for all p ≥ 1 and all 0 ≤ s < t ≤ 1, with the monotone increasing function V defined in (12). We prove it by induction. For p = 1, by independence of the N k , we have Let p ≥ 1 and assume that the property holds for p − 1. We fix 0 < s < t, and simplify the notations by setting Note that |X k | ≤ 2 for all k ≥ 1. Since (X k ) k≥1 is centered and independent, it follows that By the induction hypothesis, we infer and we deduce (16) using the monotonicity of V .
Remark 5. For the Poissonized model, we can establish similar weak convergence to the decompositions as in Theroems 1 and 2, by adapting the proofs at the end of Section 4. We omit this part.

De-Poissonization
In this section we prove our main theorems. Recall the decompositions Note that G ε andG ε , for G being Z 1 , Z 2 , U 1 , U 2 respectively, are coupled in the sense that they are defined on the same probability space as functionals of the same ε and (Y n ) n≥1 . We have already established weak convergence results forZ ε 1 ,Z ε 2 ,Ũ ε 1 ,Ũ ε 2 . The de-Poissonization step thus consists of controlling the distance between G ε andG ε . We first prove the easier part.
4.1. The processes Z ε 2 and U ε 2 . Theorem 4. For a Rademacher sequence ε, in probability, with G being Z 2 , U 2 respectively. We actually prove the above convergence in the almost sure sense. Observe that for all ε ∈ {−1, 1} N , Thus, the proof is completed once the following Lemma 5 is proved. Proof. By triangular inequality, for all n ≥ 1, t ≥ 0, First, note that for all k ≥ 1, and thus, which is bounded (since V (n)/n → 0 as n → ∞). We thus deduce (24). The proof for (25) is similar and omitted.
These identities do not hold for the process Z ε 1 or U ε 1 but we can still couple Z ε 1 ,Z ε 1 and The proof is now decomposed into two lemmas treating separately the two terms in the right-hand side of the preceding identities.
In view of Lemma 5, the following lemma will be sufficient to conclude. |q k (nt) −q k (nλ n (t))| = 0 in probability.