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.


1. 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, $\sum E[X\_n] < \infty$ $\iff$ $\sum \alpha\_n < \infty$.


2. Second question:



> 
> Is it possible that these three conditions simultaneously occur:
> 
> 
> * $\sum X\_n = +\infty$ almost surely?
> * $\sum X\_n^2 < \infty$ almost surely?
> * $\sum(\mathbb{E}[X\_n])^2 = \infty$?
> 
> 
> 



Asymptotic behavior of $X_n$ in a Dirichlet vector $(X_1, ..., X_n)$
The answer to the second question is no too. 


Set $D\_k^n=\dfrac{Y\_k}{\sum\_{i=1}^nY\_i}$. The *neutrality property* of Dirichlet vectors says that $D\_n^n=:X\_n$ is independent of $(D\_1^{n-1}, \ldots, D\_{n-1}^{n-1})$. But $X\_k=\dfrac{D\_k^{n-1}}{D\_1^{n-1}+\ldots+D\_k^{n-1}}$ for every $k=1, \ldots, n-1$, therefore $(X\_n)$ is a sequence of bounded independent random variables.


Then, by a variant of Borel-Cantelli's lemma, $\sum X\_n^2 < \infty$ $\iff$ $\sum E(X\_n^2)<\infty$. But $E(X\_n^2)\geq E(X\_n)^2$, hence the second and third bullets are opposite.


