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For any fixed n≥1, consider the randomly weighted sum ∑k=1nθkXk and the maximum max1≤m≤n∑k=1mθkXk, where Xk, 1≤k≤n, are n real-valued and not necessarily identically distributed random variables (r.v.s) with dominated variation, and θk, 1≤k≤n, are n nonnegative r.v.s without any dependence assumptions. Let Xk, 1≤k≤n, be independent of θk, 1≤k≤n. Under some relatively weaker conditions on the weights...
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