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We say that a sequence of random scalars $\{z_{n}\}$ converge in probability to a constant $\alpha$ , if for any $\epsilon>0$ ,$$\lim_{n\rightarrow\infty}\mathbb{P}\left(|z_{n}-\alpha|>\epsilon\right)=0$$

I understand the application of this notion to the Weak Law of Large Numbers, where I understand that when the sample size grows, the sampling distribution of the sample mean approaches the population value. However, the sample mean is a statistic. How can I think of a *sequence *of random scalars when it is not a statistic? What does $n$ mean in that case? Does it still mean sample size? What would convergence in probability mean in that setting? What 'distribution' or hypothetical distribution do we have in mind to compare to $\alpha$?

ChinG
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  • In other words, what does a sequence mean here? I see no reason why the $nth$ random variable will converge to anything in probability more so than the first one, if these are independent draws. What is the probability distribution over? – ChinG Sep 05 '16 at 18:23
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    When the distribution itself has n in it think $X_n$=1 with prob 1/n, 0 with prob 1-1/n, then as n changes, the random sequence converges to a value. – VCG Sep 05 '16 at 18:49
  • Given that the random sequence is in fact a random function for each step of the sequence, do we think each point coverges to a value? – ChinG Sep 05 '16 at 19:32

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