When the mean of a non-negative, integer-valued random variable goes to zero, does this imply anything about the other raw moments? When the mean of a non-negative, integer-valued random variable $X_n$, for example counting paths in a random graph on $n$ vertices between two fixed vertices, goes to zero,
$$\lim_{n \to \infty}\mathbb{E}[X_n] =0$$
can we have $\lim_{n \to \infty}\mathbb{E}[X_n^3] = \infty$, or under what conditions does this occur? This seems reasonable given we just need two different sequences to have different limits, but a bit counter-intuitive, since we're counting objects, and the raw moments give the typical number of ordered pairs, triples, etc.

If $\mathbb{E}(X_n)\to0 $ then it is totally possible for $\mathbb{E}\left(X_n^3\right)\to\infty$ as $n\to\infty$. The $X_n$'s may have finite mean but their third moments may be infinite, for example. In such a case, the limit of the third moments will be infinity (aka does not exist).
Even in the case that $X_n$'s are random variables supported on a finite set, $\mathbb{E}(X_n)\to 0$ does not imply convergence to zero of any higher moments.
