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

Michael J. Klass (), Deli Li () and Andrew Rosalsky ()
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Michael J. Klass: University of California Berkeley
Deli Li: Lakehead University
Andrew Rosalsky: University of Florida

Journal of Theoretical Probability, 2022, vol. 35, issue 3, 1556-1573

Abstract: Abstract 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 .

Keywords: Convergence; Divergence; Random series; Sums of I.I.D. random variables; Real separable Banach space; Primary 60F15; Secondary 60B12; 60G50 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10959-021-01101-9

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