Suppose $X_1, X_2, …$ are independent non-negative random variables. Show that $\sum_{n=1}^{\infty}X_n<\infty$ almost surely if and only if $\sum_{n=1}^{\infty}\mathbb E[\frac{X_n}{1+X_n}]<\infty$.
There is a hint here but I don't know how to use: Consider the truncated variables $X'_n=\min(X_n ,1)$.
My thoughts are: if the sum of expectation are finite, then the expectation should converge to zero which means $X_n$ should converge to zero almost surely and since they are independent, the sum should converges almost surely. For the other direction, maybe I should use Kolmogrov's 3-series theorem? I don't know whether this is correct. Thank you!
Best Answer
Indeed, the three series theorem helps. We can show that the finiteness of the series $\sum_{n=1}^{+\infty}\mathbb E\left[X_n/\left(1+X_n\right )\right]$ is equivalent to the convergence of the following series $$S_1:=\sum_{n=1}^{ +\infty}\mathbb P\{X_n\gt 1\},\quad S_2:=\sum_{n=1}^{ +\infty}\mathbb E\left[X_n\mathbf 1\left\{ X_n\leqslant 1\right\}\right] \mbox{ and } S_3:=\sum_{n=1}^{ +\infty}\mathbb E\left[X_n^2\mathbf 1\left\{ X_n\leqslant 1\right\}\right].$$
For $S_1$, note that $\mathbf 1\left\{ X_n\gt 1\right\}/2\leqslant X_n/(1+X_n)$ because the map $x\mapsto x/(1+x)$ is increasing on $\mathbb R_+$. For $S_2$ and $S_3$, use the bounds $$X_n^2\mathbf 1\left\{ X_n\leqslant 1\right\}\leqslant X_n\mathbf 1\left\{ X_n\leqslant 1\right\}\leqslant 2\frac{X_n}{1+X_n}.$$