Weak law of large numbers for randomly indexed sequences of m-dependent random variables

Abstract

First, we establish the inequalities related to the upper bound for the probability of the sum of a random number of random variables satisfying certain conditions. More specifically, in Theorem 1, these variables are assumed that get values on a bounded interval and in particular, are setting under m-dependence assumption instead of the usual independence, where independence is merely the specific case of m-dependence when m equal to 0. For a random index with a familiar distribution, it is possible to proceed to make reasonable estimates for the expected terms on the right-hand side of the two inequalities in Theorem 1 to obtain Chernoff-Hoeffding-style bounds. Those bounds will be employed to prove that there is a weak law of large numbers for the sequence of m-dependent random variables correspondingly and the convergence rate is exponential. Next, in Theorem 2, we had chosen the Poisson distributed index as a typical for presentation. Finally, this theorem is illustrated through an image which is constructed by simulated values of 1-dependent variables. Here, the way that we have applied to create a 1-dependent sequence from an independent sequence that it is likely will help readers understand more about m-dependence structure.