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Homework 6

  1. Suppose $ \{X_n\})_{n=1}^\infty$ is a sequence of independent r.v.s each of which is uniformly distributed on the interval $ (0,1)$ . Define a sequence of r.v.s $ \{Z_n\}$ by $ Z_n = n(1-Y_n)$ , where $ Y_N = \max_{1\leq i \leq n} X_i$ . Show that $ \{Z_n\}_{n=1}^\infty$ converges in distribution to an exponential r.v. with p.d.f.

    \begin{displaymath}f(x) =
\begin{cases}
e^{-x} & x \geq 0  0 &\text{otherwise}.
\end{cases}\end{displaymath}

  2. Suppose $ X_n \rightarrow X$ (i.p.) and that there is a constant $ C$ such that $ \vert X_n\vert \leq C$ for all $ n$ . Show that $ X_n \rightarrow X$ (m.s.)
  3. Suppose $ X_n \rightarrow C$ (in distribution), where $ C$ is a constant. Show that $ X_n \rightarrow C$ (i.p.)
Problems from Grimmet & Stirzaker
  1. Exercise 7.2.1(b,c). On the converse, suppose $ X_n$ takes values $ \pm 1$ with probability 1/2.
  2. Exercise 7.5.1.

Copyright 2008, Todd Moon. Cite/attribute Resource . admin. (2006, June 13). Homework Assignments. Retrieved January 07, 2011, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/hw6.html. This work is licensed under a Creative Commons License Creative Commons License