Random Means Generated by Random Variables: Expectation and Limit Theorems

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 Theories in Markov Dependent Random Variables

Random Variables , Distributions and Expectation 1 Random Variables

We’ve used probablity to model a variety of experiments, games, and tests. Throughout, we have tried to compute probabilities of events. We asked, for example, what is the probability of the event that you win the Monty Hall game? What is the probability of the event that it rains, given that the weatherman carried his umbrella today? What is the probability of the event that you have a rare di...

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 nonlinear hierarchical sequences of random variables

We study central limit theorems for certain nonlinear sequences of random variables. In particular, we prove the central limit theorems for the bounded conductivity of the random resistor networks on hierarchical lattices