Consider a large number of Bernoulli trials $X_1,\ldots,X_n$, where trial $X_k$ has a probability of success $\sigma_k$. So with $\sigma_k$ probability $X_k = 1$ and otherwise $X_k=0$. Further, denote $S_n = X_1 + \ldots + X_n$ - their sum. Random variable $S_n$ is distributed according to Poisson Binomial, it has expectation $\mathrm{E}[S_n]=\sigma_1 + \ldots + \sigma_n$.
I want to use the following function of their sum: $f(S_n) = \min (s, S_n)$, where $s>0$ is just some parameter. Are there any conditions when $f(S_n)$ converges to $f(\mathrm{E}[S_n])$?
Thank you in advance.