The Law of Large Numbers for $D[0,1]$-Valued Random Variables
R. Ranga Rao · Theory of Probability and Its Applications · 1963
Let $\xi _1 (t,w),\xi _2 (t,w), \ldots $ be a strictly stationary sequence of random variables taking values in the space $D[0,1]$ of real functions on $[0,1]$ without discontinuities of the second kind, and let \[ S_n (t,w) = \frac{1} {n}\left[ {\xi _1 (t,w) + \ldots + \xi _n (t,w)} \right]. \] It is proved that, for a random function $m(t,w)$ whose form is given explicitly,\[ \mathop {\lim }\limits_{n \to \infty } \left\|S_n (t,w) - m(t,w)\right\| = 0 \]with probability 1 (Theorem 1), where $\| \cdot \|$ denotes the uniform norm on $D[0,1]$. Moreover, if ${\bf E}{\| \xi _1 (t,w)\| }^{1 + \alpha } < \infty $ for some $\alpha \geqq \infty $, then\[ \mathop {\lim }\limits_{n \to \infty } {\bf E}{\left\| S_n (t,w) - m(t,w)\right\|}^{t + \alpha } = 0 \](Theorem 2).