Let $X_1,X_2,...$ be i.i.d variables with value in $\mathbb{N}$ (not necessarily finitely supported). Suppose $E(X_1) < \infty$.
Denote :
$$R_n = \textbf{Card}\{X_1,...,X_n\}$$
I must prove that $E(R_n) = o(\sqrt{n})$, but do not really know how to proceed. Does anyone have a hint :)? Many thanks !
Denote $p_k=P(X=k)$. Then $E(R_n)=\sum_k P(k\in \{X_1,\ldots,X_n\})\leqslant \sum_k \min(1,np_k)$. We are given that $\sum kp_k<\infty$. Fix $\varepsilon>0$. The sum of $\min(1,np_k)$ over $k<\varepsilon \sqrt{n}$ is of course at most $1+\varepsilon \sqrt{n}$. The sum over $k\geqslant \varepsilon \sqrt{n}$ is at most $$n\sum_{k\geqslant \varepsilon \sqrt{n}}p_k\leqslant \frac{\sqrt{n}}{\varepsilon}\sum_{k\geqslant \varepsilon \sqrt{n}}kp_k=o(\sqrt{n}).$$ Thus the result.
