Conditions on Poisson random variables to convergence in probability

Question

Let $X_1,X_2,...$ denote iid random variables such that $X_j$ has a Poisson distribution with mean $\lambda t_j$ where $\lambda$ > 0 and $t_1, t_2,...$are known positive constants.

a)Find conditions on $t_1, t_2,...$so that $\Large Y_n = \frac{\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j - \lambda }{\operatorname{Var}(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)}$ converges in distribution to a standard normal variable.

b) Suppose that, for each $j = 1,...,t_j$ lies in the interval $(a,b)$ where $0 < a < b < \infty$. Does it follow that $Y_n$ converges in distribution to a standard normal variable?

c) Suppose that $t_j$ = j, j = 1,... Does it follow that $Y_n$ converges in distribution to a standard normal variable?

Attempt at a):

Since the characteristic function of a Poisson distribution with mean $\lambda $ is given by:

$\hspace{15mm} \exp (\lambda[\exp(it)-1])$,

we then have the following characteristic function of $Y_n$:

$\hspace{15mm}$$\phi_n(t) = \exp ((\operatorname{Var}(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)^2\lambda[\exp(it/\operatorname{Var}(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j))-(\operatorname{Var}(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j))^2 - \operatorname{Var}(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)it]$

By Lemma 2.1 in Severini's "Elements of Distribution Theory":

$\exp(it) = \sum\limits_{j=0}^n \frac{(it)^j}{j!} + R_n(t)$

where

$\hspace{15mm}|R_n(t)|\leq \min(|t|^{n+1}/(n+1)!, 2|t|^n /n!$

Hence,

$\hspace{15mm}exp(it/Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)) = 1 + it/Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j) - 1/2t^2/Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)^2 + R_2(t)$

where

$\hspace{15mm} |R_2(t)|\leq 1/6t^3/Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)^{2/3}$.

It follows that

$\hspace{15mm}\phi_n(t) = exp(-t^2/2 + Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)^2 R_2(t)$

and that

$\hspace{15mm} \lim_{ Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)\to \infty}$ $Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)R_2(t) = 0$, $-\infty < t < \infty$.

Hence,

$\hspace{15mm}\lim_{ Var(\sum\limits_{j=1}^n X_j /\sum\limits_{j=1}^n t_j)\to \infty}$$\phi_n(t) = exp(-t^2/2), -\infty < t < \infty$,the characteristic function of a standard normal variable.

However, I can not identify the constraints on $t_1,t_2,....$

Any help would be much appreciated!

Answer 1

You do not follow the elementary indications given to you in comments hence it is a bit difficult to know what could help you (and the bounty is no substitute for that). Anyway, steps towards the solution could be as follows.

• The correct statement of the exercise involves $$Y_n = \frac{S_n/\tau_n - \lambda }{\sqrt{\operatorname{Var}\left(S_n/\tau_n\right)}},$$ where $$S_n=\sum\limits_{j=1}^n X_j,\qquad\tau_n=\sum\limits_{j=1}^n t_j,$$ not what is written in the question.
• The sum of independent Poisson random variables is Poisson with parameter the sum of the parameters hence $S_n$ is Poisson with parameter $\lambda \tau_n$.
• The variance of a Poisson random variable is its parameter hence $\operatorname{Var}(S_n)=\lambda \tau_n$.
• Finally, the variance of a multiple of a random variable is the variance of the random variable times the square of the multiple.

All this yields the identity $$Y_n=\frac{S_n-\lambda \tau_n}{\sqrt{\lambda\tau_n}},$$ where, for every $n$, $S_n$ is Poisson with parameter $\lambda \tau_n$. From there, everything works, for example the characteristic function approach you seem to have in mind.

Thus, the necessary and sufficient condition for the convergence in (a) to occur (which also answers (b) and (c)) is $$\tau_n\to\infty.$$