swapping limit and series

Question

Consider i.i.d non negative integer valued random variables $Z_t$ with $Y_t= \sum_{k=1}^t Z_k$ and the partial sum $$s_n:=\sum\limits_{j=c}^{\infty} P(Y_{j+1-c}=j) F^{n+c-j} 1_{\{j \leq c+n\}}$$ with

$$F^{n+c-j}:= E\left[\left(1- \frac{Y_{n+c-j}}{n+c-j}\right)\cdot 1_{\{Y_{n+c-j} \leq n+c-j \}}\right] $$

I know $$\lim\limits_{n \rightarrow \infty} F^{n+c-j}=1-z$$ for $z<1$ a.s.

I want to prove: $$\lim\limits_{n \rightarrow \infty} s_n = (1-z) \sum\limits_{j=c}^{\infty} P(Y_{j+1-c}=j)$$

Answer 1

Normally, Dominated Convergence Theorem would work fine in your case. Thing is I'm still concerned about the convergence of your limit object, especially the convergence of the series $$ \sum_{j=c}^{\infty} \mathbb{P}( Y_{m+1-c} =m ) $$ (Though the presence of that convergence is not important to for our limit proof)

Let : $$G_j(n) = F^{n+c-j} 1_{ j \le n+c}$$

According to the definition $G_m$, $0 \le G_j \le 1$ and $\lim_m G_m= (1-z) $ pointwise.

Also, for the sake of clarity, we define another object $\mu$ which is a measure on $\mathbb{N}$ such that: $$\mu(\{m\}) = \mathbb{P}( Y_{j+1-c} =j ) \text{ when }m \ge c$$ So : $$ s_n= \int G_m d\mu$$ and $$ \text{your limit object is }\int \lim G_m \mu $$

Now, let's treat your problem in different cases:

• If $\mu( \mathbb{N})<+\infty$, then $(1-z) \in L^1(\mu)$ hence by Dominated Convergence Theorem, we imply our desired result.
• If $\mu( \mathbb{N})= +\infty$, then by Fatou we have: $$\liminf s_n \ge \int \liminf G_m d\mu =+\infty $$ and note that $(1-z)\mu( \mathbb{N})$ is also equal to $+\infty$, so we still have our conclusion.
  

Discussion It'd be nice if you can expend my knowledge by having me informed of the convergence of the initial series when $\mathbb{E}(Z_1^2)=+\infty$