STAT
A Probability Path.pdf

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

Info icon This preview shows pages 212–216. Sign up to view the full content.

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.
Image of page 212

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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.
Image of page 213
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
Image of page 214

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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.
Image of page 215
Image of page 216
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern