Does the Poisson limit theorem talk about random variables or distributions?

Question

I am confused about the way we take a limit below when saying a Poisson is a limit of binomial, and also whether we are talking about random variables in the limit, or distributions.

The textbook says that the probability distribution of $X$ converges to the Poisson distribution, making me confused as if $n$ grows, we comprehend different random variables with different values of $n$; but this makes it sound as if $n$ grows, we have the same random value $X$ throughout. Here is the passage I am reading:

Theorem 19.6. Let $X \sim \operatorname{Binomial}\left(n, \frac{\lambda}{n}\right)$ where $\lambda>0$ is a fixed constant. Then for every $i=0,1,2, \ldots$, $$ \mathbb{P}[X=i] \longrightarrow \frac{\lambda^i}{i !} e^{-\lambda} \quad \text { as } n \rightarrow \infty . $$ That is, the probability distribution of $X$ converges to the Poisson distribution with parameter $\lambda$.

My question is, is the following the correct way to view the statement posited above: for any value $\lambda$, the unique distribution defined by Binomial($n, \frac \lambda n $) where $n$ is any number, approaches the Poisson distribution Poisson(\lambda) in the sense that we approach the same set of values paired with the same set of probabilities?

I am unsure if the above is the correct way to view this theorem as it didn't mention random variable at all, unlike my textbook, and also I am unsure where I talk about how the set of values assumed by the distribution are the same- as it seems we need a random variable to do this; then we could say that the set of all values which the binomial random variable assumes approaches the set of all values it would assume if it were a Poisson Random variable. So is this a correct reading of this theorem?

Answer 1

I agree the notation $X \sim \text{Binomial}(n, \lambda/n)$ can lead to confusion. You could consider a sequence of random variables $X_1, X_2, \ldots$ such that $X_n \sim \text{Binomial}(n, \lambda/n)$. Then for each $i$, you have a sequence of numbers $P(X_1=i), P(X_2=i), P(X_3=i),\ldots$ and the theorem states what the limit of this sequence is.

More broadly, this is a statement of convergence in distribution, so it is not even necessary to bring random variables into the discussion. The binomial distribution $\text{Binomial}(n, \lambda / n)$ is a distribution on the nonnegative integers (although it gives zero probability to values larger than $n$). Convergence of this sequence of distributions reduces to checking that the probability assigned to each integer $i$ converges to some number.