Abstract
Let E be a strict (LB)-space, i.e., a strict inductive limit of separable Banach spaces E 1 ⊂ E 2 ⊂ … One of the results proved in this is the following. Let X n , n ⩾ 1 be a sequence of independent identically distributed (i.i.d.) random variables taking values in E. If the Strong law of large numbers holds for this sequence, i.e., (1/ m)∑ i = 1 m X i , m ⩾ 1 converges almost surely, then there exists n ⩾ 1 such that each X i takes values in E n almost surely.
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