I want to approximate the sum
$$\sum_{k=0}^\infty \frac{\lambda^k}{k!}b_k$$
where $b_k, k \in \mathbb{N}_0$ is a bounded sequence. I was thinking about constructing a independent random variables $X_i$ with $\mathbb{E}[X_i] = \sum_{k=0}^\infty \frac{\lambda^k}{k!}b_k$ and then using the weak law of large numbers. I'm stuck at this point and don't know how to start the construction.