Some Limit Theorems for Random Functions. I

Abstract

This paper establishes a number of limit theorems for random functions $\mathcal{H}(\Delta )$ in the interval $\Delta $, which satisfy the “strong mixing condition” \[ \mathop {\lim }\limits_{\tau \to \infty } \mathop {\sup }\limits_t \mathop {\sup }\limits_{A \in M_\infty ^t ,B \in M_{t + \tau }^\infty } \left| {{\bf P}(AB) - {\bf P}(A){\bf P}(B)} \right| = 0 \] where ${\mathfrak{M}_{t_1 }^{t_2 } }$ is a $\sigma $-algebra generated by events $\{ \mathcal{H}(\Delta _1 ) < h_1 , \cdots ,\mathcal{H}(\Delta _n ) < h_n \} ,\Delta _1 , \cdots ,\Delta _n $ are arbitrary intervals, $\Delta _k \subseteq (t_1 ,t_2 ),h_1 , \cdots h_n $ are arbitrary real numbers.