18. Let Zn have the geometric distribution with parameter /n, where is fixed. Show that Zn/n converges in distribution as n , and

18. Let Zn have the geometric distribution with parameter λ/n, where λ is fixed. Show that Zn/n converges in distribution as n → ∞, and find the limiting distribution.

* 19. Let (Xk : k = 1, 2, . . . ) and (Yk : k = 1, 2, . . . ) be two sequences of independent random variables with P(Xk = 1) = P(Xk = −1) = 1 2

and P(Yk = 1) = P(Yk = −1) =

1 2



1 −

1 k2



, P(Yk = k + 1) = P(Yk = −k − 1) =

1 2k2

, for k = 1, 2, . . . . Let Sn =

Xn k=1 Xk

√n

, Tn =

Xn k=1 Yk

√n

, and let Z denote a normally distributed random variable with mean 0 and variance 1.

Prove or disprove the following: