We just covered the Central Limit theorem in class, and I stumbled upon the following reasoning that makes me think I am missing some key part of… well, something. So, here goes:
Let $X_1$ and $X_2$ be independent Poisson random variables with parameters $\lambda _1$ and $\lambda _2$ respectively. Then, $X_1 + X_2 \sim P(\lambda_1 + \lambda_2)$. By induction, for any finite $n$, if $X_i \sim P(\frac1n)$ iid, $\sum ^n X_i= X \sim P(1)$, and this holds (I think) for $n \to \infty$.
But, by the CLT,
$$ \frac{\sum ^n X_i – n\cdot\frac1n}{\sqrt{n\cdot \frac1n}}\to \tilde Y \sim N(0, 1) $$
And rearranging I get $X \to Y \sim N(1, 1)$.
But $X$ was a Poisson random variable, therefore discrete, and thus its CDF is constant everywhere except over the integers; while $Y$ is a Normal, and therefore its CDF is never constant. How is one supposed to converge to the other?
This result is clearly wrong, but I can't understand why. Can anyone please explain where in my argument I made a mistake?
Best Answer
The central limit theorem does not apply here because the distribution of the random variable you're considering depends on $n$. If you wanted to apply CLT you'd do it to $X_i \sim P(\lambda)$ and keep $\lambda$ fixed.
The relevant theorem in this situation is closer to the Poisson limit theorem.