A central limit theorem for correlated variables with limited normal or gamma distributions

Abstract

Non-negative limited normal or gamma distributed random variables are commonly used to model physical phenomenon such as the concentration of compounds within gaseous clouds. This paper demonstrates that when a collection of random variables with limited normal or gamma distributions represents a stationary process for which the underlying variables have exponentially decreasing correlations, then a central limit theorem applies to the correlated random variables.

Keywords:

• Limited normal
• gamma
• central limit
• Toeplitz matrix

AMS CLASSIFICATION:

• 60
• 62

Acknowledgements

The author wishes to acknowledge Dr. Nathan Platt and Dr. Michael Ambroso, both of the Institute for Defense Analyses, for suggesting the topic and providing insight to the associated physical phenomena.

Notes

1 See Appendix A for proof of ${\sigma }_n\to \infty$ as $n$ increases for sufficiently large ν. Also note that $E(|X_n^{2+\delta }|)<\infty$ for limited normal variates.

2 Since $\frac{\Gamma (\alpha +k)}{\Gamma (\alpha )}={\prod }_{j=0}^{k-1}(\alpha +j)$, $\alpha \ge 1\Rightarrow \frac{\Gamma (\alpha )}{\Gamma (\alpha +k)}\le \frac{1}{k!}$ and ₀$F{}_1(\alpha ,x)\le e^{2\sqrt{x}}$.

Additional information

Funding

This research was funded through central research by the Institute for Defense Analyses.