Infinite sum of random variables is infinite

Question

I am trying to better understand this statement and the assumptions made:

If $X_1,X_2,\ldots$ are non-negative independently and identically distributed random variables with $P(X_i>0)>0$, then $\displaystyle P\left(\sum_{i=1}^\infty X_i=\infty\right)=1$.

Could someone provide simple examples showing that the statement would not hold if: 1) the $X_i$ were identically distributed but not independent and 2) the $X_i$ were independent but not identically distributed? Thanks!

Answer 1

1. Let $X$ be an arbitrary non-negative random variable such that $\mathbb{P}(X>0) \in (0,1)$. Define $$X_i := X \qquad \text{for all} \, \, i \in \mathbb{N}.$$ Then the sequence is identically distributed (but not independent) and $$\mathbb{P} \left( \sum_{i=1}^{\infty} X_i = \infty \right) = \mathbb{P}(X>0) \in (0,1).$$
2. The sequence $(X_i)_{i \in \mathbb{N}}$ defined by $$X_i = \frac{1}{i^2}$$ is a sequence of independent random variables and $$\mathbb{P} \left( \sum_{i=1}^{\infty} X_i = \infty \right)=0.$$