I am encountering the following problem concerning existence of a limiting random variable (in distribution): assume a sequence of positive random variables $\{X_n\}_{n\geq 0}$ from which we know their moments ($\mathbb{E}[X_n^s]$), and also the limit of these moments (which we denote by $\{m_s\}_{\geq 0}$). Let me say that all these limits are finite numbers (no divergence).

We know that this sequence satisfies Carleman's criterion (namely, $\sum_{s\geq 0} m_s^{-1/(2s)}$ diverges), but the thing that we do NOT know in general is if this sequence defines a proper random variable $X$.

Is it clear that if this sequence satisfies the conditions of the Stieltjes moment problem (see here) then this sequence $\{X_n\}_{n\geq 0}$ converges to the corresponding random variable $X$ defined by the sequence $\{m_s\}_{\geq 0}$, and by Carleman's criterion $X$ is uniquely determined. In particular the sequence $X_n$ tends in distribution to $X$.

My question now is the following: it is possible that the sequence $\{m_s\}_{s\geq 0}$ does NOT define a random variable (namely, Stieltjes condition is NOT satisfied) but nevertheless there is convergence in distribution of my sequence to a certain random variable $X$?

In our particular case, we would like to prove that there is NOT convergence in distribution towards a Poisson (this is a very specific problem on random graphs), but we do not know to justify the last step (as in general, it is possible that there is not convergence of moments, but convergence in distribution could be true).

Thanks!