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Let ₁ (t, w), ₂ (t, w), be a strictly stationary sequence of random variables taking values in the space D0, 1 of real functions on 0, 1 without discontinuities of the second kind, and let \ Sₙ (t, w) = 1 n[ ₁ (t, w) + + ₙ (t, w). \] It is proved that, for a random function m (t, w) whose form is given explicitly, \ ₍ \|Sₙ (t, w) - m (t, w) \| = 0 probability 1 (Theorem 1), where \| \| denotes the uniform norm on D0, 1. Moreover, if E\| ₁ (t, w) \| ^1 + < for some, then\ ₍ E\| Sₙ (t, w) - m (t, w) \|^t + = 0 \ (Theorem 2).
R. Ranga Rao (Tue,) studied this question.
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