A weak law for normed weighted sums of random elements in rademacher type p banach spaces

• Published in: Journal of multivariate analysis Vol. 37; no. 2; pp. 259 - 268
• Main Authors: Adler, André, Rosalsky, Andrew, Taylor, Robert L
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.05.1991; Elsevier
• Series: Journal of Multivariate Analysis
• Subjects: convergence in probability; Exact sciences and technology; independent random elements; Mathematics; normed weighted sums; Probability and statistics; Probability theory and stochastic processes; Probability theory on algebraic and topological structures; real separable Rademacher type p Banach space; real separable Rademacher type p Banach space independent random elements normed weighted sums weak law of large numbers convergence in probability stochastically dominated random elements; Sciences and techniques of general use; stochastically dominated random elements; weak law of large numbers; independent random elements; 60B12; 60B11; real separable Rademacher type p Banach space; convergence in probability; weak law of large numbers; normed weighted sums; stochastically dominated random elements; 60F05; Weighting; Independence; Ordering; Sequence; Mean value; Banach space; Random variable
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/0047-259X(91)90083-E

For weighted sums Σ j = 1 n a j V j of independent random elements { V n , n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form ( Σ j = 1 na jV j − v n) b n → p 0 is established, where { v n , n ≥ 1} and b n → ∞ are suitable sequences. It is assumed that { V n , n ≥ 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of | V| and the growth behaviors of the constants { a n , n ≥ 1} and { b n , n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the { V n , n >- 1}.