weighted variance very short question

Hi there,

I am interested in the asymptotic variance of the weighted sample mean and weighted sample variance.

weighted_mean=sum(w(i)x(i)) if one assumes that sum(w(i))=1 and its variance is given by sigma^2*sum(w(i)^2) and is direct result of the central limit theorem.

But what about the weighted sample variance? What is its asymptotic variance?

Thank you in advance for any answers.