Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

Limit theorems for empirical processes

Summary The empirical measure P _n for iid sampling on a distribution P is formed by placing mass n ^–1 at each of the first n observations. Generalizations of the classical Glivenko-Cantelli theorem for empirical measures have been proved by Vapnik and ervonenkis using combinatorial methods. They found simple conditions on a class C to ensure that sup {|P _n (C) – P(C)|: C C} converges in probability to zero. They used a randomization device that reduced the problem to finding exponential bounds on the tails of a hypergeometric distribution. In this paper an alternative randomization is proposed. The role of the hypergeometric distribution is thereby taken over by the binomial distribution, for which the elementary Bernstein inequalities provide exponential boundson the tails. This leads to easier proofs of both the basic results of Vapnik-ervonenkis and the extensions due to Steele. A similar simplification is made in the proof of Dudley's central limit theorem forn ^1/2(P P _n –P)— a result that generalizes Donsker's functional central limit theorem for empirical distribution functions.This research was supported in part by the Air Force Office of Scientific Research, Contract No. F49620-79-C-0164     

Functional limit theorems for cumulative processes and stopping times

Summary Consider a cumulative regenerative process with increments between regeneration points being i.i.d. r.v.'s. Let the d.f. of those increments belong to the domain of attraction of a stable distribution with exponent less than two. A functional limit theorem in the Skorohod M ₁-topology is proved for this process. The M ₁-topology is more useful than the J ₁-topology in this case, because it allows the cumulative process to be continuous.The second part of the paper concerns a stopping time process, (t)--inf(s>0:w(s)>tg(s)), where w(t) is a process with positive drift for which a functional limit theorem holds and g(t)=t ^p L(t) with 0p<1 and L(t) varying slowly at infinity. Weak convergence for the process (t) is proved under certain conditions in the J ₁- and M ₁-topologies.     

Constraints on the angular distribution of the zeros of a polynomial of low complexity

LetP(z) be a polynomial of degreen with complex coefficients. Theorem.There exists a constant C such that if P has at most k terms then the number of zeros of P in any open sector of aperture π/n at the origin is no more than C ^k2. The main point of this bound is that it is independent both ofn and the coefficients ofP. The proof is a simple application of Khovanskii's real Bezout theorem to the system of real equations ReP=ImP=0. We also describe a measure of additive complexity for sets of integers and use it to estimate the angular distribution more finely.     

Extremes in autoregressive processes with uniform marginal distributions

Chernick (1981) derives a limit theorem for the maximum term for a class of first order autoregressive processes with uniform marginal distributions. The parameter for these processes is equal to 1/r where r is an integer, r 2. Based on this limit theorem, the asymptotic distribution of the minimum term and the joint asymptotic distribution of the maximum and minimum terms in the sequence are obtained. Since the condition D′(u_n) of Leadbetter (1974) fails, the condition of Davis (1979), D′(v_n, u_n), also fails. Negatively correlated uniform sequences are shown to exist. Asymptotic distributions for the maximum and minimum terms in the sequence are derived and it is shown that the maximum and minimum are not asymptotically independent.     

Limit Theorems for Supercritical Branching Random Fields

An infinite particle system in R^d is considered where the initial distribution is POISSON ian and each initial particle gives rise to a supercritical age-dependent branching process with the particles moving randomly in space. Our approach differs from the usual: instead of the point measures determined by the locations of the particles at each time, we take the particles at a “final time” and observe the past histories of their ancestry lines. A law of large numbers and a central limit theorem are proved under a space-time scaling representing high density of particles and small mean particle lifetime. The fluctuation limit is a generalized GAUSS -MARKOV process with continuous trajectories and satisfies a deterministic evolution equation with generalized random initial condition. A more precise form of the central limit theorem is obtained in the case of particles performing BROWN ian motion and having exponentially distributed lifetime.     

A Functional Central Limit Theorem for Positively Dependent Random Variables

In this note we prove a functional central limit theorem for LPQD processes, satisfying some assumptions on the covariances and the moment condition sup_j≥1E|X_j|^2+ρ < ∞ for some ρ > 0.     

Some central limit theorems for ℓ∞-valued semimartingales and their applications

Summary. This paper is devoted to the generalization of central limit theorems for empirical processes to several types of ℓ^∞(Ψ)-valued continuous-time stochastic processes t⇝X _t ⁿ =(X _t ^{n ,ψ}|ψ∈Ψ), where Ψ is a non-empty set. We deal with three kinds of situations as follows. Each coordinate process t⇝X _t ^{n ,ψ} is: (i) a general semimartingale; (ii) a stochastic integral of a predictable function with respect to an integer-valued random measure; (iii) a continuous local martingale. Some applications to statistical inference problems are also presented. We prove the functional asymptotic normality of generalized Nelson-Aalen's estimator in the multiplicative intensity model for marked point processes. Its asymptotic efficiency in the sense of convolution theorem is also shown. The asymptotic behavior of log-likelihood ratio random fields of certain continuous semimartingales is derived. Received: 6 May 1996 / In revised form: 4 February 1997     

Functional central limit theorem for super α-stable processes

A functional central limit theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α < d < 2α (0 < α ≤ 2), the limiting process is a Gaussian process, whose covariance is specified; for the critical dimension d= 2α and higher dimensions d < 2α, the limiting process is Brownian motion. Zhang Mei, Functional central limit theorem for the super-brownian motion with super-Brownian immigration, J. Theoret. Probab., to appear.     

Minimal extensions of Π01 classes

A minimal extension of a Π⁰₁ class P is a Π⁰₁ class Q such that P ? Q, Q – P is infinite, and for any Π⁰₁ class R, if P ? R ? Q, then either R – P is finite or Q – R is finite; Q is a nontrivial minimal extension of P if in addition P and Q′ have the same Cantor‐Bendixson derivative. We show that for any class P which has a single limit point A, and that point of degree ≤ 0 , P admits a nontrivial minimal extension. We also show that as long as P is infinite, then P does not admit any decidable nontrivial minimal extension Q. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functional central limit theorems for a large network in which customers join the shortest of several queues

We consider N single server infinite buffer queues with service rate . Customers arrive at rate N, choose L queues uniformly, and join the shortest. We study the processes for large N, where R^N_t(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data R^N₀ satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the R^N in equilibrium under the stability condition <, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits lim _N lim _t= lim _t lim _N by a compactness-uniqueness method. We deduce a posteriori the CLT for R^N₀ under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit.Mathematics Subject Classification (2000):Primary: 60K35; Secondary: 60K25, 60B12, 60F05, 37C75, 37A30     

The Maximum of a Critical Branching Wiener Process

Consider a critical branching Wiener process on ¹. Let M(n) be the location of the most right particle at time n. A limit distribution theorem is proved for n ^–1/2 M(n).     

Central limit theorems for nonlinear functionals of stationary Gaussian processes

Let X=(X _t ,t) be a stationary Gaussian process on (, ,P), letH(X) be the Hilbert space of variables inL ² (,P) which are measurable with respect toX, and let (U _s ,s) be the associated family of time-shift operators. We sayYH(X) (withE(Y)=0) satisfies the functional central limit theorem or FCLT [respectively, the central limit theorem of CLT if in [respectively,], where

A central-limit-theorem version ofL=λw

Underlying the fundamental queueing formulaL=W is a relation between cumulative processes in continuous time (the integral of the queue length process) and in discrete time (the sum of the waiting times of successive customers). Except for remainder terms which usually are asymptotically negligible, each cumulative process is a random time-transformation of the other. As a consequence, in addition to the familiar relation between the with-prob ability-one limits of the averages, roughly speaking, the customer-average wait obeys a central limit theorem if and only if the time-average queue length obeys a central limit theorem, in which case both averages, properly normalized, converge in distribution jointly, and the individual limiting distributions are simply related. This relation between the central limit theorems is conveniently expressed in terms of functional central limit theorems, using the continuous mapping theorem and related arguments. The central limit theorems can be applied to compare the asymptotic efficiency of different estimators of queueing parameters. For example, when the arrival rate is known and the interarrivai times and waiting times are negatively correlated, it is more asymptotically efficient to estimate the long-run time-average queue lengthL indirectly by the sample-average of the waiting times, invokingL=W, than it is to estimate it by the sample-average of the queue length. This variance-reduction principle extends a corresponding result for the standard GI/G/s model established by Carson and Law [2].Supported by the National Science Foundation under Grant No. ECS-8404809 and by the U.S. Army under Contract No. DAAG29-80-C-0041.     

Gaussian fluctuations of connectivities in the subcritical regime of percolation

Summary We consider thed-dimensional Bernoulli bond percolation model and prove the following results for allp _c : (1) The leading power-law correction to exponential decay of the connectivity function between the origin and the point (L, 0, ..., 0) isL ^–(d–1)/2 . (2) The correlation length, (p) is real analytic. (3) Conditioned on the existence of a path between the origin and the point (L, 0, ..., 0), the hitting distribution of the cluster in the intermediate planes,x ₁ =qL,0, obeys a multidimensional local limit theorem. Furthermore, for the two-dimensional percolation system, we prove the absence of a roughening transition: For allp>p _c , the finite-volume conditional measures, defined by requiring the existence of a dual path between opposing faces of the boundary, converge—in the infinite-volume limit—to the standard Bernoulli measure.Work supported in part by G.N.A.F.A. (C.N.R.)Work supported in part by NSF Grant No. DMS-88-06552     

Semi-parametric Estimation of the Hölder Exponent of a Stationary Gaussian Process with Minimax Rates

Let (X(t))_t∈[0,1] be a centered Gaussian process with stationary increments such that IE[(X_{u+t-X_u)²] = C|t|^s+r(t). Assume that there exists an extra parameter β > 0 and a polynomial P of degree smaller than s + β such that |r(t)-P(t)| is bounded with respect to |t|^s+β. We consider the problem of estimating the parameter s ∈ (0,2) in the asymptotic framework given by n equispaced observations in [0, 1]. Adding possibly stronger regularity conditions to r, we define classes of such processes over which we show that s cannot be estimated at a better rate than n^{min(1/2, β)}. Then, we study increment (or, more generally, discrete variation) estimators. We obtained precise bounds of the bias of the variance which show that the bias mainly depend on the parameter β and the variance on two terms, one depending on the parameter s and one on some regularity properties of r. A central limit theorem is given when the variance term relying on s dominates the bias and the other variance term. Eventually, we exhibit an estimator which achieves the minimax rate over a wide range of classes for which sufficient regularity conditions are assumed on r. This revised version was published online in August 2006 with corrections to the Cover Date.     

Joint functional convergence of partial sums and maxima for linear processes<Superscript>*</Superscript>

For linear processes with independent identically distributed innovations that are regularly varying with tail index α ∈ (0, 2), we study the functional convergence of the joint partial-sum and partial-maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes, which enable the functional convergence in the space of ?²-valued càdlàg functions on [0, 1] with the Skorokhod weak M₂ topology.We also obtain a joint convergence in the M₂ topology on the first coordinate and in theM₁ topology on the second coordinate.     

Combinatorial decompositions and homogeneous geometrical processes

This paper considers line processes and random mosaics. The processes are assumed invariant with respect to the group of translations ofR ². An expression for the probabilities ,k=0, 1, 2,... to havek hits on an interval of lengtht taken on a typical line of direction (the hits are produced by other lines of the process) is obtained. Also, the distribution of a length of a typical edge having direction in terms of the process {P _i , _i } is found, hereP _i is the point process of intersections of edges of the mosaic with a fixed line of direction and the mark _i is the intersection angle atP _i . The method is based on the results of combinatorial integral geometry.     

On asymptotics of the potential of a countable ergodic Markov chain

For a class of functionsf, the convergence in Abel's sense is proved for the potential _no P ⁿ f(i) of a uniform ergodic Markov chain in a countable phase space. Several corollaries are obtained which are useful from the point of view of the possible application to CLT (the central limit theorem) for Markov chains. In particular, we establish the condition equivalent to the boundedness of the second moment for the time of the first return into the state.Translated from Ukrainskii Matematicheskii Zhumal, Vol. 45, No. 2, pp. 265–269, February, 1993.     

A FUNCTIONAL CENTRAL LIMIT THEOREMFOR NEGATIVELY ASSOCIATED SEQUENCE

Let(x1,j≥1)be a sequence of negatively associated random variables with ex1=o,ex^21＜∞.in this paper a functional central limit theorem for negatively associated random variables under some conditions withbout stationarity is proved which is the same as the results for positively associated random variables.