2
$\begingroup$

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 !

$\endgroup$

1 Answer 1

5
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.