522, Statistics Problem Set 2 Wellner; 1/23/2008
Reading: Shorack, PfS, Shorack, PfS, Due:
Chapter 11, Sections 7-8, pages 312 - 318; Chapter 13, Sections 1-7, pages 367 - 390. Wednesday, January 30, 2008.
1. (Symmetry and conditional expectation). Let X1 , X2 , . . . be i.i.d. random variables with the same distribution as X where E|X| < . Let Sn X1 + + Xn , and dene Gn [Sn , Sn+1 , . . .] = [Sn , Xn+1 , Xn+2 , . . .] . Show that E(X1 |Sn ) = n1 Sn almost surely. [Hint: Note that [Xn+1 , Xn+2 , . . .] is independent of [X1 , Sn ], and use symmetry to show that E(1[Sn B] X1 ) = E(1[SnB] X2 ) = = E(1[Sn B] Xn ).] 2. (An application of the WLLN: Inversion of Laplace transforms) Let P be a probability measure on the Borel subsets of [0, ), and dene its Laplace transform by (t) = 0 etx dP (x) for t [0, Widders ). inversion formula for P from is:
[nz] n
lim
k=0
(1)k k (k) n (n) = P ([0, z)) k!
(1)
Show that (1) holds via the following steps: (a) Dierentiation of the integral k times shows that (k) (t) =
0
(x)k etx dP (x).
(b) Setting t = n, letting z > 0, multiplying across by (1)k nk /k!, and summing on k yields
[nz]
k=0
(1)k k (k) n (n) = k! 1
[nz] 0 k=0
enx
(nx)k dP (x) k!
(2)
where enx (nx)k /k! = P (Sn = k) and Sn = Y1 + +Yn where Y1 , Y2, . . . are i.i.d. Poisson...