Law of large numbers for the function of sum of random variables

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 $f(\mathrm{E}[S_n])$?

• Saying a sequence converges to another sequence is meaningless; what did you really mean to ask? And did you really mean to ask about $S_n$, as opposed to $S_n/n$? – David C. Ullrich Aug 15 '15 at 13:43
• Not sure what 'Poisson-Binomial' means. Not sure what any of it means. Total speculation, for what it may happen to be worth: If all of the $\sigma_i$ (horrible notation, do you really mean $\theta$?) then $\sum_{i=1}^\infty \sigma_i$ diverges to $\infty.$ Then $E(S_n)$ also goes to infinity and $f(E[S_n)] \rightarrow s$. Also, $P(S_n \ge s) = P(f(S_n) = s) \rightarrow 1.$ And you have your convergence. Maybe the condition is that $\sum_{i=1}^\infty \sigma_i$ diverges. – BruceET Aug 15 '15 at 22:45