Let $(X_i)_{i=1}^\infty$ be independent nonnegative integer valued random variables. Suppose that $X_n \succeq X_{n+1}$ (in the stochastic dominance sense). Does it follow that $X_n \overset{d}\to X$ for some random variable $X$?
It seems to me that if we let $F_n(k) = \mathbf P[ X_n \leq k]$, then because $X_n \succeq X_{n+1}$, we have $F_n(k)$ are increasing, and have limits, $F(k)$. I believe the stochastic dominance also implies that $F(k) \leq F(k+1) \leq 1$ for all $k$. So, it appears $F$ defines a unique distribution for a r.v. $X$. Is this reasoning sound?
