The central limit theorem for stationary Markov processes with normal generator—with applications to hypergroups

We extend the central limit theorem for additive functionals of a stationary, ergodic Markov chain with normal transition operator due to Gordin and Lif?ic, 1981 [A remark about a Markov process with normal transition operator, In: Third Vilnius Conference on Probability and Statistics 1, pp. 147–48] to continuous-time Markov processes with normal generators. As examples, we discuss random walks on compact commutative hypergroups as well as certain random walks on non-commutative, compact groups.     

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.     

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.     

Kahane–Khintchine inequalities and functional central limit theorem for stationary random fields

We establish new Kahane–Khintchine inequalities in Orlicz spaces induced by exponential Young functions for stationary real random fields which are bounded or satisfy some finite exponential moment condition. Next, we give sufficient conditions for partial sum processes indexed by classes of sets satisfying some metric entropy condition to converge in distribution to a set-indexed Brownian motion. Moreover, the class of random fields that we study includes φ-mixing and martingale difference random fields.     

The invariance principle for ϕ-mixing sequences

Summary In this paper we investigate the invariance principle for -mixing sequences, satisfying restrictions on the variances which are a weak form of stationarity. No mixing rate is assumed. For -mixing strictly stationary sequences we give a necessary and sufficient condition for the invariance principle.     

Rates of uniform convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions

Assume that {Xn} is a strictly stationary β-mixing random sequence with the β-mixing coefficient βk = O(k-r), 0 < r ≤1. Yu (1994) obtained convergence rates of empirical processes of strictly stationary β-mixing random sequence indexed by bounded classes of functions. Here, a new truncation method is proposed and used to study the convergence for empirical processes of strictly stationary β-mixing sequences indexed by an unbounded class of functions. The research results show that if the envelope of the index class of functions is in Lp, p > 2 or p > 4, uniform convergence rates of empirical processes of strictly stationary β-mixing random sequence over the index classes can reach O((nr/(l+r)/logn)-1/2) or O((nr/(1+r)/ log n)-3/4) and that the Central Limit Theorem does not always hold for the empirical processes.``     

The central limit theorem for Chébli-Trimèche hypergroups

Let (S _nn>-1) be a random walk on a hypergroup ( ₊ , *), i.e., a Markov chain with transition kernelN(x, A) = _x * (A), where is a fixed probability measure on ₊ such that the second moment exists. Then depending on the growth of the hypergroup two situations can occur: when ( ₊ , *) is of exponential growth then it is shown thatS _n is asymptotically normal. In the case of polynomial growth {more precisely, if the densityA of the Haar measure of ( ₊ , *) satisfies lim[A()/A()]=}, the normalized variablesS _n/[n Var()/(+1)]^1/2 converge to a Rayleigh distribution with parameter .     

Berry–Esseen bounds in the entropic central limit theorem

Berry–Esseen-type bounds for total variation and relative entropy distances to the normal law are established for the sums of non-i.i.d. random variables.     

On the invariance principle for stationary ϕ-mixing triangular arrays with infinitely divisible limits

Summary Given a stationary, -mixing triangular array of Banach space valued random vectors whose row sums converge weakly to an infinitely divisible probability measure, necessary and sufficient conditions for the validity of the corresponding invariance principle in distribution are given.     

On the central limit theorem in R p when p→∞

Summary Let X ₁ , X ₂ , ..., X _n be i.i.d. random vectors in R ^p where p tends to infinity. A theorem is presented showing that the Central Limit Theorem should hold if p ²/n tends to zero. Furthermore, an example is presented with X _i having a mixed multivariate normal distribution (with finite moment generating function) for which a uniform normal approximation to the distribution of the sample mean can not hold if p ²/n does not tend to zero. Research supported in part by National Science Foundation Grants MCS 80-02340, MCS 83-01834, and DMS 85-03785     

A central limit theorem for Ramanujan’s tau function

A central limit theorem is established for the absolute value of the modular Fourier-coefficient function defined by Ramanujan, and for that of the error term in the formula counting representations of integers as sums of twenty-four squares, in which the function appears.     

Complete q-order moment convergence of moving average processes under φ-mixing assumptions

Let {Y _i , ?∞ < i < ∞} be a doubly infinite sequence of identically distributed φ-mixing random variables, and {a _i , ?∞ < i < ∞} an absolutely summable sequence of real numbers. We prove the complete q-order moment convergence for the partial sums of moving average processes $\left\{ {X_n = \sum\limits_{i = - \infty }^\infty {a_i + Y_{i + n} ,n \geqslant 1} } \right\}$ based on the sequence {Y _i , ?∞ < i < ∞} of φ-mixing random variables under some suitable conditions. These results generalize and complement earlier results.     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ^?1 Σ _k ^=0n?1 Xk for a stationary processX = (X _n ,n ∈ ?) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n ¦σ _k ¦ ≥ε) asn → ∞ are obtained.     

Designs of ϕ-optimal control for second-order processes

Consider a realization of the process on the intervalT=[0,1] for functionsf ₁(t),f ₂(t),...,f _n (t) inH(R), the reproducing kernel Hilbert space with reproducing kernelR(s,t) onT×T, whereR(s,t)=E[ξ(s)ξt)] is assumed to be continuous and known. Problems of the selection of functions {f _k (t)} _k=1 ⁿ to be ϕ-optimal design are given, and an unified approach to the solutions ofD-,A-,E- andD _s-optimal design problems are discussed.     

Analogues of the central point theorem for families with d-intersection property in ℝ d

In this paper we consider families of compact convex sets in ? ^d such that any subfamily of size at most d has a nonempty intersection. We prove some analogues of the central point theorem and Tverberg’s theorem for such families.