Interpretation when the limit of a sequence of random variables go to infinitive. I would like to ask for what exactly means that:
$\lim_{x\to \infty} X_n$ where $X_n$ is a sequence of random variables.
Can be interpreted it like that $\lim_{x\to \infty} X_n= \cup_{i=1}^\infty X_n$?
This question comes in the context of convergence modes for random variables.
 A: Random variables $X_i : \Omega \to \mathbb R$ are functions from the sample space $\Omega$ to the real numbers (they can also be defined onto the complex, or other euclidean spaces), which need to be measurable (you can ignore this if you aren't familiar). Intuitively, these functions assign quantitative values to each outcome of a random experiment. For example, the age of a randomly selected person from planet earth is a random variable.
There are many ways to define limits for random variables, as there are many ways to define limits of functions. Some of the most common are:

*

*Pointwise. In this case, $X_n \to X$ if, for every outcome $\omega \in \Omega$, we have $X_n(\omega)$ (a sequence of real numbers) converging to $X(\omega)$ (a real number).

*Almost surely. The same as before, but the convergence can fail in a subset of $\Omega$ of probability $0$.

*In probability/measure. We require that the probability that $X_n$ and $X$ are far away goes to zero. In other words, that $P(|X_n - X| > \epsilon)$ (a sequence of real numbers) tends to $0$ (a number) for every $\epsilon > 0$. This is weaker than the previous notions (can you see why?).

*In distribution. This is an even weaker notion that, loosely, requires that $P(X_n \le t)$ (a sequence of real numbers) converges to $P(X \le t)$ (a number) for every $t \in \mathbb R$ (actually, some pathological values of $t$ are exempted).

*In ($L^p$) norm This is defined for $p \in [1,\infty)$, and requires that $\int_\Omega|X_n - X|^pdP$ (a sequence of real numbers) converges to zero. (Similarly it can be defined for other values of $p$ but for simplicity I stick to these).

All of these notions capture different behaviors of the sequence $X_i$ and some of the most important theorems of probability theory (Central Limit, Large Numbers...) are devoted to showing some of these convergences for certain sequences. Hopefully with this overview you can search for further information.
