STAT
A Probability Path.pdf

# Define for n 2 1 sn 27 1 x and suppose oo and a s p

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Define for n 2:: 1, Sn = 2::7 = 1 X; and suppose = oo and a; :S p C/ln forsomec > Oandalln.ShowSn/E(Sn)--+ 1.

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6. 7 Exercises 199 23. A classical transform result says the following: Suppose Un 0 and Un -. u as n -. oo. For 0 < s < 1, define the generating function 00 U(s) = n=O Show that lim(l- s)U(s) = u S-+1 by the following relatively painless method which uses convergence in probability: Let T (s) be a geometric random variable satisfying P[T(s) = n] = (1- s)sn. Then T(s) oo. What is E(ur(s))? 24. Recall a random vector (Xn, Yn) (that is, a random element of JR 2 ) con- verges in probability to a limit random vector (X, Y) if p d((Xn, Yn), (X, Y)) -. 0 where d is the Euclidean metric on JR 2 . (a) Prove (6.23) iff p p X n __. X and Yn __. Y. (b) Iff : JR 2 ..... JRd is continuous (d 1), (6 . 23) implies p /(Xn, Yn)-. /(X, Y). (c) If (6.23) holds, then 25. For random variables X, Y define p(X, Y) = inf{8 > 0: P[IX- Yi 8) ::::: 8}. (a) Show p(X, Y) = 0 iff P[X = Y] = 1. Form equivalence classes of random variables which are equal almost surely and show that p is a metric on the space of such equivalence classes. (b) This metric metrizes convergence in probability: Xn iff p(Xn, X)-. 0.
200 6. Convergence Concepts (c) The metric is complete: Every Cauchy sequence is convergent. (d) It is impossible to metrize almost sure convergence. 26 . Let the probability space be the Lebesgue interval; that is, the unit interval with Lebesgue measure. (a) Define n Xn = - 1 -leo n-t>, n 3. ogn · Then {Xn} is ui, E(Xn)-. 0 but there is no integrable Y which dom- inates {Xnl· (b) Define Xn = nlco.n-1)- nlcn-t,2n-l)· Then {Xn} is not ui but Xn 2:. 0 and E(Xn)-. 0. 27. Let X be a random variable in L1 and consider the map x : [1 , oo] 1-4 [0, oo] defined by x(p) = IIXIIp · Let Po:= sup{p 1: IIXIIp < oo}. Show x is continuous on [1, po). Furthermore on [1, po) the continuous function p 1-4 log IIXIIp is convex. 28. Suppose u is a continuous and increasing mapping of [0, oo] onto [0, oo]. Let u <-- be its inverse function. Define for x 0 U(x) =fox u(s)ds, V(x) =fox u<-(s)ds. Show xy:::: U(x) + V(y), x, y E [0, oo]. (Draw some pictures.) Hence, for two random variables X, Y on the same probability space, XY is integrable if U(IXI) E L1 and V(IYI) E L1. Specialize to the case where u(x) = xP- 1 , for p > 1. 29. Suppose the probability space is ((0, 1], B((O, 1]), A) where A is Lebesgue measure. Define the interval where 2P + q = n is the decomposition of n such that p and q are integers satisfying p 0, 0 :::: q < 2P. Show lAn 2:. 0 but that lim sup lAn = 1, n->oo lim inf lAn = 0. n->oo

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6. 7 Exercises 201 30. The space L 00 : For a random variable X define IIXIIoo = sup{x : P[IXI > x] > 0}. Let L 00 be the set of all random variables X for which IIXIIoo < oo . (a) Show that for a random variable X and 1 < p < q < oo 0 IIXIIt IIXIIp IIXIIq IIXIIoo · (b) For 1 < p < q < oo, show L 00 CLqCLpCLI. (c) Show Holder's inequality holds in the form E(IXYI):;: IIXIIt IIYIIoo · (d) Show Minkowski's inequality holds in the form IIX + Ylloo IIXIIoo + IIYIIoo· 31. Recall the definition of median from Exercise 31 of Chapter 5.
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