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?

  • 4
    $\begingroup$ 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 Oct 12 '16 at 13:00
  • $\begingroup$ Possible duplicate of Convergence of Sequence Random Variables $\endgroup$ – Xi'an Oct 12 '16 at 13:13
  • $\begingroup$ @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 Oct 12 '16 at 13:40
  • $\begingroup$ 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 Oct 12 '16 at 14:31

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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