Sum of non i.i.d. Bernoulli and Le Cam's theorem

Question

Consider a (deterministic) sequence $(p_i)_i$ such that $p_i \in (0, 1)$ and $\sum_{I}^n p_i = T_n \rightarrow T < +\infty$.

Then define a sequence of independent Bernoulli random variables $B_i \sim \text{Be}(p_i)$.

From le Cam's theorem, I know that $S_n = B_1 + \cdots + B_n$ is approximately Poisson distributed with parameter $T_n$. Is there a result that shows that $S_{\infty} = \lim_{n \rightarrow \infty} S_n$ is Poisson distributed with parameter $T$? I know of similar limit results when $T_n$ is constant across $n$ (see Chen, 1974 at https://www.jstor.org/stable/2959300), but I'm missing something with this more general setting.

EDIT: I have found some places where the result I'm looking for is stated (e.g., https://d-nb.info/1197139125/34, https://stats.stackexchange.com/questions/391461/limit-behavior-of-weighted-poisson-binomial-distribution) but no proof is given.

Looking at Le Cam's statement, it seems that the upper bound of the total variation distance between the law of $S_n$ and the Poisson distribution could be large if some of the $p_i$'s do not tend to zero.

Answer 1

The main difference between le Cam's theorem and the setting of the question is that here, the parameter $p_i$ is only allowed to depend on $i$, not on $n$.

The limiting random variable, namely, $Y=\sum_{i\geqslant 1}B_i$, cannot be Poisson distributed, since $$ \mathbb E[Y]=T, \operatorname{Var}(Y)=\sum_{i\geqslant 1}\operatorname{Var}(B_i)=T-\sum_{i\geqslant 1}p_i^2. $$