Let X1,X2, . . . , Xn be i.i.d. Gaussian with mean and variance 2. (a)

Let X1,X2, . . . , Xn be i.i.d. Gaussian with mean μ and variance σ2.

(a) Show that the mean and variance of Y1 = |X1 − X2| are σ4

π and 2σ2 1 − 2

π  respectively.

(b) Prove that the covariance between Y1 = |X1 − X2| and Y2 = |X2 − X3|

is 2σ2 1 6

− (2−

√

3)

π

. This exercise is difficult.

(c) Consider the estimation of σ given by

ˆσn =

π

4

× 1 n − 1 n



i=2

|Xi − Xi−1|.

Deduce from

(a) and

(b) that E (ˆσn) = σ and nVar (ˆσn) → 2π

3 +

√

3−3 ≈

.8264, as n→∞.