A Central Limit Theorem for Additive Random Functions

Yu. A. Rozanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 5 (1960) no. 2, pp. 243-246

Résumé

In this paper the additive random functions ${\text{H}}(\Delta )$ of the semi-interval $\Delta=[s,t)$, satisfying the strong mixing condition (1), are considered. Let in formula (1) the variable $\alpha(\tau)= O[\tau^{-1-\varepsilon}]$ and $\mathbf M|\mathrm H(\Delta_0)-\mathbf M\mathrm H(\Delta_0)|^{2+\delta}\leq M_0,\delta>2/\varepsilon$ for all $\Delta_0=[t_0,t+t_0)$, then, assuming condition (4), $$\mathbf P\biggl\{\frac{\mathrm H(\Delta)-\mathbf M\mathrm H(\Delta)}{\sqrt{\mathbf D\mathrm H(\Delta)}} x\biggr\}\to\frac1{\sqrt {2\pi}}\int_{-\infty}^x{e^{-u^2/2}}\,du$$ when $|\Delta|=t-s\to \infty$.