# Poisson process on Skorokhod's space

For each $$n=1,2,\ldots$$, let $$\ \xi_{n1},\ldots, \xi_{nn}$$ be random and independent variables such that $$\mathbb{P}(\xi_{ni}=1)=p_n \ \$$ and $$\ \ \mathbb{P}(\xi_{ni}=0)=1-p_n$$. Let consider the process: $$Y_n(t)=\sum_{k=1}^{[nt]}\xi_{nk} \ \ \ \ \ t\in [0,1],$$ where $$[x]$$ represents the floor of $$x$$.

Supposing that $$np_n\to \lambda\$$ as $$n$$ goes to $$\infty$$, with $$\lambda>0$$, prove that: $$Y_n \to Y \ \ \ (\text{in distribution), in} \ D[0,1],$$ where $$Y$$ is a Poisson process with parameter $$\lambda$$, and $$D[0,1]$$ is the Skorokhod's (Skorohod's) space.

So what I got is: $$\xi_{ni}\in Be(p_n),$$ for all $$n$$ and for all $$i=1,\ldots ,n$$.
And with this and the independence of $$\xi_{n1},\ldots ,\xi_{nn}$$, we deduce that: $$Y_n(t)=\sum_{k=1}^{[nt]}\xi_{nk} \in Bin([nt],p_n)$$ I also know that if $$X_n\in Bin(n,p_n) \ \$$ and $$\ \ np_n \to \lambda$$, then $$\ \ X_n\to Poisson(\lambda) \ \$$(in distirbution.)

I dont't know how to continue with this. I am also not sure if I can use that: $$[nt]p_n \to \lambda t \ \ \text{,and then,} \ \ Y_n(t)\to Poisson(\lambda t)$$ I am really lost from here since even if what I said was true, that doesn't implies it happens in $$D[0,1]$$.

Any help will be appreciated.

$$\def\la{\lambda}$$ $$\def\To{\Rightarrow}$$ $$\def\bR{\mathbb{R}}$$ $$\def\al{\alpha}$$ $$\def\be{\beta}$$ $$\def\ep{\varepsilon}$$

Yes, you can use what you wrote to conclude that for each fixed $$t$$, the sequence of random variables $$Y_n(t)$$ converges in distribution to a Poisson with parameter $$\la t$$. Or in other words, $$Y_n(t) \To Y(t)$$ in $$\bR$$.

This, however, is not enough to show that $$Y_n\To Y$$ in $$D[0,1]$$. For this, you will want to show two things:

1. For every finite set $$\{t_1,\ldots,t_k\}\subset[0,1]$$, $$(Y_n(t_1), \ldots, Y_n(t_k)) \To (Y(t_1), \ldots, Y(t_k))$$ in $$\bR^k$$, and
2. The sequence $$\{Y_n\}$$ is relatively compact in $$D[0,1]$$.

In any proof of convergence of this type, (2) tends to be the most difficult. Things are sometimes a little easier in $$C[0,1]$$, but in $$D[0,1]$$, here is a sufficient condition for relative compactness:

There exists $$\al,\be,C>0$$ such that for all $$t$$ and $$\ep$$ satisfying $$0\le t-\ep, we have $$E[|Y_n(t+\ep) - Y_n(t)|^\al|Y_n(t) - Y_n(t-\ep)|^\al] \le C\ep^{1+\be}$$ for all $$n$$.

This sufficient condition is an immediate consequence of Theorem 3.8.8 in Ethier and Kurtz.