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.
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Adler, A., Rosalsky, A. & Volodin, A.I. Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces. Journal of Theoretical Probability 10, 605–623 (1997). https://doi.org/10.1023/A:1022645526197
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DOI: https://doi.org/10.1023/A:1022645526197