Divergence Criterion for a Class of Random Series Related to the Partial Sums of I.I.D. Random Variables

Let $$ \{X, X_{n};~n \ge 1 \}$$ { X , X n ; n ≥ 1 } be a sequence of independent and identically distributed Banach space valued random variables. This paper is devoted to providing a divergence criterion for a class of random series of the form $$\sum _{n=1}^{\infty } f_{n}\left( \left\| S_{n} \right\| \right) $$ ∑ n = 1 ∞ f n S n where $$S_{n} = X_{1} + \cdots + X_{n}, ~n \ge 1$$ S n = X 1 + ⋯ + X n , n ≥ 1 and $$\left\{ f_{n}(\cdot ); n \ge 1 \right\} $$ f n ( · ) ; n ≥ 1 is a sequence of nonnegative nondecreasing functions defined on $$[0, \infty )$$ [ 0 , ∞ ) . More specifically, it is shown that (i) the above random series diverges almost surely if $$\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) = \infty $$ ∑ n = 1 ∞ f n c n 1 / 2 = ∞ for some $$c > 0$$ c > 0 and (ii) the above random series converges almost surely if $$\sum _{n=1}^{\infty } f_{n} \left( cn^{1/2} \right) 0$$ c > 0 provided additional conditions are imposed involving X, the sequences $$\left\{ S_{n};~n \ge 1 \right\} $$ S n ; n ≥ 1 and $$\left\{ f_{n}(\cdot ); n \ge 1 \right\} $$ f n ( · ) ; n ≥ 1 , and c. A special case of this criterion is a divergence/convergence criterion for the random series $$\sum _{n=1}^{\infty } a_{n} \left\| S_{n} \right\| ^{q}$$ ∑ n = 1 ∞ a n S n q based on the series $$\sum _{n=1}^{\infty } a_{n} n^{q/2}$$ ∑ n = 1 ∞ a n n q / 2 where $$\left\{ a_{n};~n \ge 1 \right\} $$ a n ; n ≥ 1 is a sequence of nonnegative numbers and $$q > 0$$ q > 0 .