Hello,
I wish to calculate (or upper bound) expectations of the form $E[\langle x,y \rangle^2]$, where $x$ and $y$ are i.i.d standard gaussian vectors of length n. Are there any exponential type upper bounds for the same?
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1$\begingroup$ WHat is an "exponential-type upper bound"? $\endgroup$– Igor RivinCommented Nov 25, 2011 at 21:51
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$\begingroup$ an upper bound that exponentially decays to 0 $\endgroup$– user19530Commented Nov 25, 2011 at 22:36
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$\begingroup$ I think I am misunderstanding the question along the same lines as Igor Rivin's answer below... another way of phrasing his answer is: let $W_i = x_iy_i$, then since each $W_i$ has mean zero you are looking at the variance of a sum of $n$ i.i.d. copies of $W_1$, and $W_1$ has variance $1$, so the whole sum has variance $n$. $\endgroup$– Yemon ChoiCommented Nov 26, 2011 at 0:05
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1 Answer
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I must be misunderstanding the question, but $<x, y>^2$ is a sum of the terms of the form $x_i x_j y_i y_j.$ The expectation of this term vanishes, unless $i=j,$ in which case it (the . expectation) is the square of the expectation of the square of the standard Gaussian. The square of the standard gaussian is the chi-square distribution with one degree of freedom, whose mean is $1,$ so the whole thing should be $n$.
answered Nov 25, 2011 at 21:58
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$\begingroup$ Perhaps I didnt pose the question correctly. What I am looking for is something along these lines: let $\theta = <x,y>$, where x and y are as defined earlier. so $\theta$ is a sum of bessel functions, whose mean is 0. I am now looking to upper bound the expected value of $\theta^2$ . $\endgroup$ Commented Nov 25, 2011 at 22:35
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2$\begingroup$ Perhaps you can explain Igor's misunderstanding, since I have it too. $\endgroup$ Commented Nov 26, 2011 at 14:59