Is there an example of two discrete random variables $X, Y$ s. t. $\mathbb{V}(X + Y) = \infty$, but $V(X) < \infty, V(Y) < \infty$?
It's obvious that they must be dependent (that is, $\mathrm{P}(X = a \ \wedge \ Y = b) \neq P(X = a)\cdot P(Y = b)$) and that's enough to $\mathbb{E}XY = \sum_{i=1}^{\infty}\sum_{j=1}^{\infty}a_i\cdot b_j \cdot P(X = a_i \ \wedge \ Y = b_j)$ be infinite. If $P(X = a_i \ \wedge \ Y = b_j)$ is not small enough to make $\mathbb{E}XY$ converge we're done. But I can't come up with it.
Maybe, for discrete random variables $\mathbb{V}(X + Y) < \infty$ always?
Thank you.
As said earlier, you have $V(X+Y) < \infty$ iif $\mathbb{E}((X+Y)^2) < \infty$. But, because of the Cauchy-Schwartz inequality, $$\mathbb{E}((X+Y)^2) \le \mathbb{E}[X^2 + Y^2] + 2 \mathbb{E}[X^2]^{\frac12} \mathbb{E}[Y^2]^{\frac12}$$ and since, by assumption, $Var(X) < \infty$, then $\mathbb{E}[X^2] < \infty$, the same holds for $Y$ and thus $V(X+Y) < \infty$ in any case. This apply whether your variables are discrete or continuous.