Let $X_n$ and $Y_n$ be two different and independent random variables for $n = \{1,2, ...\}$. We cannot say $X_i$ and $Y_j$ are independent if $i \neq j$. The values of $X_n$ is $R$ while the values of $Y_n$ are non-negative.

We want to calculate $E[\sum\limits_{i=1}^{\infty} X_i Y_i]$.

My question is:

Can we use $E[\sum\limits_{i=1}^{\infty} X_i Y_j] = \sum\limits_{i=1}^{\infty} E[X_iY_i]$ even $X_i$ is not non-negative? I am a little confused about the condition of interchanging expectation and summation.

I tried to understand the similar question(Interchange of expectation and summation) but failed.


No, it is not true. Suppose $X_{2n}=1$ and $X_{2n+1}=-1$ with probability $1$, and $Y_i=1$ with probability $1$. Then neither series converges.


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