Weak Limit of the Geometric Sum of Independent But Not Identically Distributed Random Variables

Abstract

We show that when $\{X_j\}$ is a sequence of independent (but not necessarily identically distributed) random variables which satisfies a condition similar to the Lindeberg condition, the properly normalized geometric sum ${\sum }_{j=1}^{{\nu }_p}{X}_j$ (where ${\nu }_p$ is a geometric random variable with mean $1/p$) converges in distribution to a Laplace distribution as $p\to 0$. The same conclusion holds for the multivariate case. This theorem provides a reason for the ubiquity of the double power law in economic and financial data.

Cited by in corpus (3)

• Stein's method and the Laplace distribution
• New error bounds for Laplace approximation via Stein's method
• On convergence of the distributions of random sequences with independent random indexes to variance-mean mixtures