# On large deviations for sums of discrete m-dependent random variables

## TL;DR

This paper investigates large deviation probabilities for sums of discrete m-dependent random variables, comparing their distributions to Poisson, negative binomial, and binomial approximations under various dependence and moment conditions.

## Contribution

It provides new large deviation results for sums of m-dependent variables, extending classical approximations to dependent discrete cases with specific distribution comparisons.

## Key findings

- Ratio $P(S_n=x)/P(Z_n=x)$ analyzed for different dependence structures.
- Approximation of $P(S_n \u2265 x)$ by Poisson distribution established.
- Results applicable to sums of dependent indicators and run statistics.

## Abstract

The ratio $P(S_n=x)/P(Z_n=x)$ is investigated for three cases: (a) when $S_n$ is a sum of 1-dependent non-negative integer-valued random variables (rvs), satisfying some moment conditions, and $Z_n$ is Poisson rv; (b) when $S_n$ is a statistic of 2-runs and $Z_n$ is negative binomial rv; and (c) when $S_n$ is statistic of $N(1,1)$-events and $Z_n$ is a binomial r.v. We also consider the approximation of $P(S_n\geqslant x)$ by Poisson distribution with parameter depending on $x$.