The standard asymptotic theory would start by deriving the
characteristic funciton of |R(i)|, then of a(i)|R(i)|, then multiplying
together the characteristics functions. Then invert the characteristic
function with liberal use of Taylor's theorem.
Any good book on asymptotics and approximation theory in Statistics
(especially Edgeworth expansions) will discuss this. The modern theory
of saddlepoint approximations may do something different, but I'm not
familiar with that.
Hope this helps.
Spencer Graves
Thomas W Blackwell wrote:
If this were my problem, I would try to separate real and imaginary
parts - write everything out in polar coordinates - and recognize
chi-squared random variables where they occur. But that's very much
a beginner's approach.
HTH - tom blackwell - u michigan medical school - ann arbor -
On Mon, 14 Apr 2003, comm wrote:
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