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I am a bit confused when studying the convergence of random variables. All the material I read using $X_i, i=1:n$ to denote a sequence of random variables. Based on the theory, a random variable is a function mapping the event from the sample space to the real line, in which the outcome is a real value number. Then the $\{X_i(\omega)\}$ is a sequence of real value numbers. There is no confusion here.

But when talking about convergence of random variables, it goes to $X_n \rightarrow X$ in probability or in distribution. For example, if we toss a coin once, the sample space is $\{tail = 0, head = 1\}$ and the outcome is 0 or 1. If we toss 10 times, each time it is a random variable of outcome 0 or 1. Then no matter how big is the $n$, $X_n$ still equals to 0 or 1 from one tossing. How can we talk about the convergence of random variables from this sense? Unless $X_i$ is the toss of $i=1...n$ times in one experiment with underlying sample space $2^i$, then define a sequence of random variables the number of head counts in $i=1...n$ so that $X_n\rightarrow X$ in probability.

How should I get my head around this?

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    $\begingroup$ You took a wrong turn at the end of the first paragraph where you wrote "there is no confusion here": $(X_i)$ is a sequence of real valued functions, not numbers. For each $\omega$ in the sample space, $(X_i(\omega))$ is a sequence of real numbers. $\endgroup$
    – whuber
    Commented Oct 12, 2016 at 13:00
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    $\begingroup$ Possible duplicate of Convergence of Sequence Random Variables $\endgroup$
    – Xi'an
    Commented Oct 12, 2016 at 13:13
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    $\begingroup$ @whuber, I supposed to mean the sequence of the outcome. So can I understand that a sequence of random variable is a sequence of function of n? Then when $n\rightarrow \infty$, it converge to a function $X$? $\endgroup$
    – Gavin Wu
    Commented Oct 12, 2016 at 13:40
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    $\begingroup$ I think my confusion is $\{X_i\}$ is a sequence of random variables, and $\{Y_i\}$ given by $Y_n=\frac{\sum_{i=1}^n X_i}{n}$ is also a sequence of random variables. Can we talk about the convergence of $X_n$ in the same way as $Y_n$ does? $\endgroup$
    – Gavin Wu
    Commented Oct 12, 2016 at 14:31

2 Answers 2

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As I understand this. You should have some Randome Variables $X_n$ which depends on $n$.

For example, if you take a look at this post: Convergence in distribution, probability, and 2nd mean You'll find that if $n \rightarrow \infty$ then $X_n$ converges in probability. Depeding on RVs you have different types of converging.

For your example you can take $Y_n = \frac{1}{n}\sum_{k=1}^{n}X_k$ and it should converge to 0.5. You could have 10 heads in a row, but as $n \rightarrow \infty$ then $Y_n \rightarrow 0.5$

I hope I answered your question.

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The following contents are just copy-paste from: Sequence of Random Variables.

Here, we would like to discuss what we precisely mean by a sequence of random variables. Remember that, in any probability model, we have a sample space $S$ and a probability measure $P$. For simplicity, suppose that our sample space consists of a finite number of elements, i.e., $$ S=\left\{s_{1}, s_{2}, \cdots, s_{k}\right\} $$ Then, a random variable $X$ is a mapping that assigns a real number to any of the possible outcomes $s_{i}, i=1,2, \cdots, k .$ Thus, we may write $$ X\left(s_{i}\right)=x_{i}, \quad \text { for } i=1,2, \cdots, k $$ When we have a sequence of random variables $X_{1}, X_{2}, X_{3}, \cdots$, it is also useful to remember that we have an underlying sample space $S$. In particular, each $X_{n}$ is a function from $S$ to real numbers. Thus, we may write $$ X_{n}\left(s_{i}\right)=x_{n i}, \quad \text { for } i=1,2, \cdots, k $$ In sum, a sequence of random variables is in fact a sequence of functions $X_{n}: S \rightarrow \mathbb{R}$.

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