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Asymptotic behavior of row sums in 2-d array of random variables

Set-up. Let $f : \mathbb{N} \to \mathbb{N}$ be increasing. For each $m \in [0,1]$, consider an infinite two-dimensional array of random variables, where row $n$ has $f(n)$ variables: $B^m_{1,1}$ $B^m …