Let {X
k
, k=1,2,…} be a sequence of independent random variables with common subexponential distribution F, and let {w
k
, k=1,2,…} be a sequence of positive numbers. Under some mild summability conditions, we establish simple asymptotic estimates for the extreme tail probabilities of both the weighted sum ∑
k=1
n
w
k
X
k
and the maximum of weighted sums max1≤m≤n
∑
k=1
m
w
k
X
k
, subject to the requirement that they should hold uniformly for n=1,2,…. Potentially, a direct application of the result is to risk analysis, where the ruin probability is to be evaluated for a company having gross loss X
k
during the kth year, with a discount or inflation factor w
k
.