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

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?

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.