I'm trying to understand convergence in probability and have a specific problem; how do you show that the sample mean of n random variables $(X_k)$, each of which is the mean of $Y_{k-1}$ through $Y_{k+1}$ random variables $(Y_k\sim N(0,10))$, converges in probability to $0$ as $n$ approaches infinity? I think it's quite obvious that it DOES, but I don't know how to show it...
Convergence of a running average
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statistics
convergence-divergence
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0k = 1, 2, 3, ... , n – 2012-12-07
1 Answers
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Your question does not give the dependency structure of the $\{Y_k\}$. Usually, in a problem like this, the elements of $\{Y_k\}$ would be mutually independent. Assuming this is the case, then (if I understand the problem correctly) you have $X_k = (1/3)(Y_{k-1} + Y_k + Y_{k+1})$ for $k=1,...,n$. It should be clear that the sum $\sum_{k=1}^n X_k = (1/3)(\sum_{k=0}^{n-1}Y_k + \sum_{k=1}^{n}Y_k + \sum_{k=2}^{n+1}Y_k)$. You can now work out the asymptotics of each of the these three inner sums separately, and then combine the results.
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0I have edited my answer to divide by three, as you suggest. – 2012-12-07