Functional Limit Theorems for Dependent Variables

Central Limit Theorems for Dependent Random Variables

1. Limiting distributions of sums of 'small' independent random variables have been extensively studied and there is a satisfactory general theory of the subject (see e.g. the monograph of B.V. Gnedenko and A.N. Kolmogorov [2]). These results are conveniently formulated for double arrays Xn k (k = 1, . . . , kn ; n = 1, 2, . . . ) of random variables where the Xn k (k = 1, . . . , kn), the rand...

Central and Local Limit Theorems in Markov Dependent Random Variables

We consider an irreducible and aperiodic Markov chain {kn}n=0 over the finite state space E = {1, . . . , p} with positive regular transition matrix P = {pij} and additive component {Un} such that {Sn} = {(kn, Un)} is also a Markov chain over the state space E1 = E × R. We prove a central and a local limit theorem for this chain when the probability density functions of {Sn}, conditional on the...

Central and Local Limit Theories in Markov Dependent Random Variables

Functional Limit Theorems for Digital Expansions

The main purpose of this paper is to discuss the asymptotic behavior of the difference sq,k(P (n))−k(q−1)/2 where sq,k(n) denotes the sum of the first k digits in the q-ary digital expansion of n and P (x) is an integer polynomial. We prove that this difference can be approximated by a Brownian motion and obtain under special assumptions on P a Strassen’s type version of the law of the iterated...

Some Useful Functions for Functional Limit Theorems

Many useftil descriptions of stochastic models can be obtained from functional limit theorems (invariance principles or weak convergence theorems for probability meastires on function ŝpaces). These descriptions typically come from standard functional limit theorems via the o^ntinuous mapping theorem. This paper facilitates applications of the continuous mapping theorem by determining when seve...