Convergence of sum of a linear combination of Poisson variables

Let $Y_j$ with $j=1,...,m$ be independent Poisson random variables with parameter $\lambda_j$. I need some hints to find (provided that it exists, so with some condition on the sequence $\lambda_j$) the limit in distribution of the sum

$$
X_m = \sum_{j=1}^m j\,Y_j

$$

when $m\rightarrow\infty$.

Answer

Some steps:

• If $X$ is a random variable with Poisson distribution of parameter $\lambda$, then its characteristic function is given by $ \varphi(s)= \exp(\lambda(e^{is}-1))$.


• From this, we can deduce the characteristic function of $j Y_j$.


• Using independence, we get the characteristic function of $X_m$ for each $m$. Now the problem is deterministic, since we have to investigate the pointwise convergence of a sequence of functions.