 Expectation of an absolutely summable random series

If random variables $X_i$ satisfy ${\sum }_{i}E|X_i|<\infty$, then ${\sum }_i{X}_i$ converges absolutely almost surely, is integrable, and its expected value equals ${\sum }_{i}E[X_i]$. Indeed, the monotone convergence theorem gives $E[{\sum }_i|X_i|]={\sum }_{i}E|X_i|<\infty$, so that nonnegative sum is finite almost surely. It dominates every finite partial sum in absolute value; the dominated convergence theorem then justifies passing the limit through the expected value. No independence is required.

For example, if $|X_i|\le M{q}^i$ for a fixed $M$ and $0<q<1$, the geometric series bound proves the hypothesis directly. This is useful when repeated observations are multiplied by geometrically shrinking distances or weights.