Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

André Adler, Andrew Rosalsky () and Andrej I. Volodin

Journal of Theoretical Probability, 1997, vol. 10, issue 3, 605-623

Abstract: Abstract For a sequence of constants {a n,n≥1}, an array of rowwise independent and stochastically dominated random elements { V nj, j≥1, n≥1} in a real separable Rademacher type p (1≤p≤2) Banach space, and a sequence of positive integer-valued random variables {T n, n≥1}, a general weak law of large numbers of the form $$\sum {_{j = 1}^{T_n } } a_j (V_{nj} - c_{nj} )/b_{[\alpha _n ]} \xrightarrow{P}0$$ is established where {c nj, j≥1, n≥1}, α n → ∞, b n → ∞ are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V nj, j≥1, n≥1}. Illustrative examples include one wherein the strong law of large numbers fails.

Keywords: Rademacher type p Banach space; array of rowwise independent random elements; weighted sums; weak law of large numbers; random indices (search for similar items in EconPapers)
Date: 1997
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DOI: 10.1023/A:1022645526197

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