# correlations about half normal distributions

• Jan 7th 2013, 12:59 PM
GerrardWu
Hi all:

I want to bound the following quantity:

$\displaystyle f(n)=\mathbf{E}_{h_1,\ldots,h_n,y_1,\ldots,y_n}[sign(\sum_{i}h_iy_i) \prod_{i}y_i]$

where $\displaystyle h_1,\ldots,h_n$ are independent half normal random variables, and $\displaystyle y_1,\ldots,y_n$ are uniformly random {-1,1}.

For n even, f(n) is 0.

By some experiment simulation we can show that for n =4k+1, f(n)>0 ; and for n = 4k+3, f(n)<0.

My questions is can we give lower bounds for the absolute value of the quantity? i.e, can we show that |f(n)|> something inverse exponential of n?

Or even easier, can we show that for n odd, f(n) is not 0?

Thanks
• Jan 7th 2013, 03:53 PM
chiro
Re: correlations about half normal distributions
Hey GerrardWu.

Have you tried using some kind of analysis theorem on the integral of the y_i's?

Something like this:

Absolute convergence - Wikipedia, the free encyclopedia

But instead you consider a sum of logarithms and then the exponential of that sum.
• Jan 7th 2013, 08:05 PM
GerrardWu
Re: correlations about half normal distributions
Hi chiro:

Thank you very much for you quick reply, I don't quite understand what you mean by the integral of the y_i's, as here y_i are discrete random variable, which is 1 or -1 with equal probability.

Quote:

Originally Posted by chiro
Hey GerrardWu.

Have you tried using some kind of analysis theorem on the integral of the y_i's?

Something like this:

Absolute convergence - Wikipedia, the free encyclopedia

But instead you consider a sum of logarithms and then the exponential of that sum.

• Jan 7th 2013, 08:17 PM
chiro
Re: correlations about half normal distributions
Sorry I confused the y_i's with the h_i's.

Basically you can look at the expectation term of the products by consider the nth integral of 1/2^n dV where dV is over n integrals.

I'm ignoring the sign term since you want to look at the magnitude.

Have you tried looking at E[Multiply all Y_I]?