Almost sure convergence for weighted sums of pairwise PQD random variables

TL;DR

This paper proves strong laws of large numbers for weighted sums of pairwise positively quadrant dependent variables, and applies these results to confirm the consistency of certain regression estimators with dependent errors.

Contribution

It introduces new strong law results for weighted sums of pairwise PQD variables dominated by an $L_p$ variable, extending classical laws to dependent data.

Findings

Established strong laws of large numbers for weighted sums of PQD variables.

Proved strong consistency of regression estimators with PQD error terms.

Extended classical probabilistic results to dependent, non-independent variables.

Abstract

We obtain Marcinkiewicz-Zygmund strong laws of large numbers for weighted sums of pairwise positively quadrant dependent random variables stochastically dominated by a random variable $X \in \mathscr{L}_{p}$, $1 \leqslant p < 2$. We use our results to establish the strong consistency of estimators which emerge from regression models having pairwise positively quadrant dependent errors.