Determine if sequence snnbspconverges in distribution

Question

Suppose  X_k is a sequence of zero-mean Gaussian random variables with covariances described by  Form the sequence of sample means

Note that in this case we are forming the sequence of sample means of dependent random variables

(a) Determine if the sequence S_n converges in distribution.

(b) Determine if the sequence S_n  converges in probability.

(c) Determine if the sequence S_n converges in the MS sense.