Another way in which underlying assumptions can be violated is if there is correlation in the sampling, which can seriously affect the properties of the sample mean. Suppose we introduce correlation in the case discussed in Exercise 10.2.1; that is, we observe X1,..., Xn, where Xi ~ n(θ, σ2), but the Xis are no longer independent.
(a) For the equicorrelated case, that is, Corr(Xi, Xj) = p for i ‰  j, show that

so Var() †› 0 as n †’ ˆž.
(b) If the Xis are observed through time (or distance), it is sometimes assumed that the correlation decreases with time (or distance), with one specific model being Corr(Xi, Xj) = p|i - j| Show that in this case

so Var() †’ 0 as n †’ ˆž. (See Miscellanea 5.8.2 for another effect of correlation.)
(c) The correlation structure in part (b) arises in an autoregressive AR(1) model, where we assume that Xi+1 = pXi + δi, with δi iid n(0,1). If |p|

72 2 1-p