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