Abstract
For an array {V nk ,k≥1,n≥1} of rowwise independent random elements in a real separable Banach space \(X\) with almost surely convergent row sums \(S_n = \sum {_{k = 1}^\infty {\text{ }}V_{nk} ,n \geqslant 1} \), we provide criteria for S n −A n to be stochastically bounded or for the weak law of large numbers \(S_n - A_n \xrightarrow{P}0\) to hold where {A n ,n≥1} is a (nonrandom) sequence in \(X\).
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Rosalsky, A., Volodin, A.I. On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces. Journal of Theoretical Probability 17, 327–346 (2004). https://doi.org/10.1023/B:JOTP.0000020697.69668.e7
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DOI: https://doi.org/10.1023/B:JOTP.0000020697.69668.e7