The Law of Large Numbers for $D[0,1]$-Valued Random Variables

Let ξ ₁ (t,w),ξ ₂ (t,w), … 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 \[ S_n (t,w) = 1/n[ {ξ _1 (t,w) + … + ξ _n (t,w)} ]. \] It is proved that, for a random function $m(t,w)$ whose form is given explicitly,\[ {lim }n → ∞ \|S_n (t,w) - m(t,w)\| = 0 \]with probability 1 (Theorem 1), where \| · \| denotes the uniform norm on $D[0,1]$. Moreover, if E\| ξ ₁ (t,w)\| 1 + α < ∞ for some α ∞, then\[ {lim }n → ∞ { E}{\| S_n (t,w) - m(t,w)\|}t + α = 0 \](Theorem 2).