Limit theorems for the sums of random variables related to a Markov chain. I

Limit theorems for sums of random variables with mixture distribution

In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of light-tail and truncated heavy-tail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions of the sums due to various relations between these ...

Limit theorems for sums of random exponentials

Received: date / Revised version: date Abstract. We study limiting distributions of exponential sums SN (t) = ∑N i=1 e tXi as t → ∞, N → ∞, where (Xi) are i.i.d. random variables. Two cases are considered: (A) ess supXi = 0 and (B) ess supXi = ∞. We assume that the function h(x) = − log P{Xi > x} (case B) or h(x) = − log P{Xi > −1/x} (case A) is regularly varying at ∞ with index 1 < % < ∞ (case...

Central and Local Limit Theories in Markov Dependent Random Variables

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 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...