Edited after comment by Ofer Zeitouni

I have a sequence of discrete time stochastic processes $\big((S_n(i))_{i \geq 1}\big)_{n\geq 1}$ such that for every $n$, $i$,

\begin{equation}S_n(i)=\sum_{j=1}^i Y_n(j)\end{equation} where $Y_n(j)$ are non negative integer valued random variables.

I know that

\begin{equation}\lim_{n \to \infty} \mathbb E\Big[\sum_{j=1}^\infty Y_n(j)\mathbf 1_{Y_n(j)\geq 2}\Big]=0,\end{equation} that is, that the expected contribution to the process from non-unitary jumps converges to $0$, and exists a function $\lambda (t): [0,\infty)\mapsto (0,\infty)$ and a sequence $a_n$ with $a_n \to \infty$ such that, for every $k\in \mathbb N$, and every sequence of arrays $(j_{1,n},j_{2,n},...,j_{k,n})_{n \geq 1}$ such that $(j_{1,n}/a_n,j_{2,n}/a_n,...,j_{k,n}/a_n)_{n \geq 1}\to (t_1,t_2,...,t_k) \in \mathbb [0,\infty)^k$,

\begin{equation}\lim_{n \to \infty}\mathbb E[Y_n(j_{1,n})Y_n(j_{2,n})\cdots Y_n(j_{k,n})]a_n^{k}= \lambda(t_1)\lambda(t_2)\cdots \lambda (t_k).\end{equation}

Can I conclude that the continuous time process $(S_n(\lfloor a_nt\rfloor ))_{t \geq 0}$ converges in distribution to an inhomogeneous Poisson point process with intensity $\lambda (t)$ (even in some quite weak topology, like for finite dimensional distributions)?