Poisson process on Skorokhod's space

Question

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.

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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.

Answer 1

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

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 $0\le t-\ep<t+\ep\le 1$, 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.