I'll provide you some assurance that the equality in reference does in fact hold. Write everything out in summation form, expand it all out, cancel the terms that you can and things will simplify. One thing to keep in mind
it's just a very simple change to the the definition of but it's useful to keep in mind when dealing with these summation problems.
I don't have emotions and sometimes that makes me very sad.
The Chebyshev inequality is:
where . In particular we put , then we have
From your work I think you have applied it correctly. The spirit of this useful bound is that if you want to show a sequence of random variable is converging towards to its common mean, then then you just need to show the variance is converging to zero.
The tricky part here is that
1. the calculation of the variance can be quite tedious
2. the mean of is a sequence of , converging to but not exactly equal to ( is asymptotically unbiased, but biased estimator of ). So you may need to have a little adjustment before applying it.
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