Let $(X_n)$ be a sum of i.i.d. positive random variables such that $\mathbb{E}(X_1)=1$ and $\mathbb{P}(X_1\neq 1)>0$. Put $M_n=X_1\ldots X_n$. Show that $\sum _{n\geq 1}\sqrt{M_n}< +\infty $ a.e.
It can be show that $M_n$ is a martingale so that $\sqrt{M_n}$ is a supermartingale but this doesn't help. I don't see how to use the condition $\mathbb{P}(X_1\neq 1)>0$ in order to show that this serie converge.
Let $(X_n)$ be a sequence of i.i.d. positive random variables such that $\mathbb{E}(X_1)=1$ and $\mathbb{P}(X_1\neq 1)>0$.
$\mathbb{E}\big( \sqrt{M_n} \big) =\mathbb{E}\big( \sqrt{X_1\ldots X_n} \big) = \Big[ \mathbb{E}( \sqrt{X_1} ) \Big]^n =: q^n $, where $q : =\mathbb{E}( \sqrt{X_1} ) < \sqrt{\mathbb{E}(X_1)} = 1 $, note that the strict inequality exactly follows from the Jensen inequality and the assumption that $\mathbb{P}(X_1\neq 1)>0$, since the case of equality only occurs when $X_1 = 1$ a.s.
It remains to apply Fubini:
$$\mathbb{E}\left[ \sum_{n=1}^\infty \sqrt{M_n} \right] =\sum_{n=1}^\infty \mathbb{E}\big( \sqrt{M_n} \big) = \sum_{n=1}^\infty q^n < +\infty \,\, . $$
So the desired result follows. QED
