[RP]Lecture Note VII

# [RP]Lecture Note VII - Kyung Hee University Department of...

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VII-1 C1002900 RP Lecture Handout VII: Stochastic Convergence and Limit Theorems Reading: Chapter 7.4 A random or a discrete-time random process is a sequence of random variables ,, n XX X 12  (VII.1) Since a random variable is a function on for some probability space  P , a se- quence of random variables is a sequence of functions. There are many possible definitions for convergence. 7.1. Four Definitions of Convergence Almost Sure (everywhere) Convergence Definition 7.1 : A sequence of random variables   n X converges almost surely (or eve- rywhere) to a random variable X if   lim n n   1 . (VII.2) Almost sure convergence is denoted by a.s. or a.e n  . (VII.3) Conceptually, to check almost sure convergence, one can first set     :lim n n  and then see if it have probability one.

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Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VII-2 Example 7.1 : Let   n X be the sequence of random variables on the standard unit- internal probability space   ,  0 1 , defined by    ,, n n X   0 1 . (VII.4) 1 1  X 1 X 3 X 2 11 1 1 0 0 0 Figure 7.1: n n X on   , 01. This sequence converges for all  with the limit if lim if . n n X    00 1 (VII.5) Since the single point set   1 have probability zero, i.e.,   10 ,   n X converges almost surely to zero. That is, if let X be the zero random variable, defined by X 0 for all   , 01 , then a.s. n XX  . (VII.6)          :lim / /. n n    1 1  Convergence in the MS sense Definition 7.2 : A sequence of random variables   n X converges to a random variable X in the mean square sense if   n X  2 for all n and
Kyung Hee University Random Processing Department of Electronics and Radio Engineering Prof. Hyundong Shin Communications and Coding Theory Laboratory (CCTLAB) VII-3   lim n n XX   2 0 . (VII.7) Mean square convergence is denoted by m.s. n  . (VII.8) Convergence in Probability Definition 7.3 : A sequence of random variables   n X converges in probability to a random variable X if for any  0 ,   lim n n  0 . (VII.9) Convergence in probability is denoted by p. n . (VII.10) Example 7.2 : Let   n X be a sequence of random variables defined on the standard unit- internal probability space   ,  01 as  ,i f ,o t h e r w i s e n n an X   01 0 (VII.11) where n a 0 is some constant depending on n .

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## This note was uploaded on 06/10/2010 for the course ELECTRONIC C1002900 taught by Professor Hyungdongshin during the Spring '10 term at Kyung Hee.

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[RP]Lecture Note VII - Kyung Hee University Department of...

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