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 ${\mathbf E}{\|\xi_1 (t,w)\|}^{1+\alpha}<\infty$ for some $\alpha\geqq\infty$, then $$ \mathop{\lim}\limits_{n\to\infty}{\mathbf E}{\left\|S_n(t,w)-m(t,w)\right\|}^{t+\alpha}=0 $$.(Theorem 2).