# [R] A statistical problem.Anybody can help me?

comm comm at 263.net
Mon Apr 14 04:36:46 CEST 2003

```Sorry for the contents not relating to R.

Assume there are N i.i.d zero-mean complex gaussian random
variables(RVs),as w(i),0<=i<N} with known variance,from which one
can generate another N RVs,as

R(0)=sum over i {w(i)*w'(i)}
R(1)=sum over i {w(i+1)*w'(i)}
...
up to
R(N-1)= w(N-1)w'(i)

where w'(i) is the complex conjugate of w(i).
(from viewpoint of signal processing, R(i) are serial correlation of time series w(i))

If one defines a new random variable using {R(k)} as

Z=a(0)R(0)+a(1)|R(1)|+...  a(N-1)|R(N-1)|,

with {a(k)} are known and |.| is modulus operation.It's a decision
statistic encountered in my work. I wish to find its approximated(using
Central Limit Theorem) statistical characteristics in close-form.Mean and
variance are enough.

Does anybody have any ideas or references which can solve this problem?

(below is my previous thoughts and now it is tested not work because RVs appear to be Rician distributed)
Mean of Z is easy to get. However its variance is troublesome. I think it can be calculated by

Var=alpha*C*alpha',

where alpha=[a(0) a(1) ... a(N-1)],C is covariance matrix of vector [R(0),|R(1)|,...,|R(N-1)|].

Besides the diagonal and first row and first column, the other elements is
small that can be ignored,which can be shown by simulations.Namely weak
cross-correlation is hold between any two RVs of set
{|R(1)|,|R(2)|,R(N-1)},while the crosss-correlation between R(0) and each
RV of set {|R(1)|,|R(2)|, |R(N-1)|} and self-correlation of set
{|R(0)|,|R(1)|,|R(N-1)|} is large and should not be ignored. The former is
what i seek. I almost exhausted,so i came here for help.

Any suggestion will be appreciated.

Regards,
Jeans Sun

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