Let $(\alpha_k)$ be a sequence of positive numbers and let $(Y_k)$ be a sequence of independent random variables $Y_k \sim \text{Gamma}(\alpha_k,1)$. Set $X_n=\dfrac{Y_n}{\sum_{i=1}^nY_i}$.

**(edit)** The title is not appropriate: $(X_1, ..., X_n)$ is not a Dirichlet vector, because $X_k=\dfrac{Y_k}{\sum_{i=1}^kY_i} \neq \dfrac{Y_k}{\sum_{i=1}^nY_i}$. Sorry for the confusion.

- First question:

Is it possible that these two conditions simultaneously occur:

$\sum X_n < +\infty$ almost surely?

$\sum \mathbb{E}[X_n] = \infty$?

**(edit) Answer to question 1:** *No*. One has $\sum_{i=1}^nY_i =\dfrac{1}{\prod_{i=2}^n(1-X_i)}$, hence $\sum X_n < +\infty$ $\iff$ $\sum Y_n < \infty$ $\iff$ $\sum \alpha_n < \infty$ (because $\sum Y_n \sim \text{Gamma}(\sum \alpha_n,1))$. Now, $E[X_n]=\dfrac{\alpha_n}{\sum_{i=1}^n\alpha_i}$, and by a similar reasoning, $\sum E[X_n] < \infty$ $\iff$ $\sum \alpha_n < \infty$.

- Second question:

Is it possible that these three conditions simultaneously occur:

$\sum X_n = +\infty$ almost surely?

$\sum X_n^2 < \infty$ almost surely?

$\sum(\mathbb{E}[X_n])^2 = \infty$?

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