Chapter4

# Chapter4 - Stat 1301B Probability Statistics I Chapter IV...

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Stat 1301B Probability & Statistics I Fall 2011-2012 Chapter IV Limit Theorems The probabilistic behaviour of the sample mean when the sample size n is large (say, tends to infinity) is called the limiting distribution of the sample mean. Law of large number ( LLN ) and the central limit theorem ( CLT ) are two of the most important theorems in statistics concerning the limiting distribution of the sample mean. These two theorems suggest the “nice” properties of the sample mean and justify its advantages. Before proceeding, we need to define what ‘convergence’ means in the context of random variables. § 4.1 Modes of Convergence Let be a sequence of random variables (not necessarily independent), X be another random variable. Let ,... , 2 1 X X ( ) x F X ( ) x F n X be the distribution function of , n X be the distribution function of X . Converges in Distribution / Converges in Law / Weak Convergence n X is said to converge in distribution to if X ( ) ( ) x F x F X X n n = lim for all points x at which is continuous. It is denoted as . () x F X X X L n ⎯→ Example 4.1 Let . Define as the maximum of . Then the distribution function of is given by ( 1 , 0 ~ ,... , 2 1 U U U iid ) n X n U U U ,..., , 2 1 n X 0 = x F n X for 0 x ; 1 = x F n X for 1 x ; () ( ) ( ) x U x U x U P x X P x F n n X n = = ,..., , 2 1 ( ) ( ) x U P x U P x U P n = L 2 1 n x = , for 1 0 < < x . P.165

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Stat 1301B Probability & Statistics I Fall 2011-2012 Therefore () < = 1 if 1 1 if 0 lim x x x F n X n . On the other hand, consider a random variable X which is degenerated at 1, i.e. . The distribution function of X is 1 1 = = X P () ( ) < = = 1 if 1 1 if 0 x x x X P x F X . Hence () () x F x F X X n n = lim and thereby . We may also write X X L n ⎯→ 1 L n X as X is degenerated at 1. Now suppose we define ( n n X n Y ) = 1 , then the distribution function of is n Y ( ) ()() = = = = n y F n y X P y X n P y Y P y F n n X n n n Y 1 1 1 1 < < = n y n y n y y n if 1 0 if 1 1 0 if 0 . Therefore < < = y e y y F y Y n n 0 if 1 0 if 0 lim which is the distribution function of ( ) 1 Exp . Hence ( ) n n X n Y = 1 1 = converges in distribution to an exponential random variable with parameter λ , i.e. ( ) ( ) 1 ,..., , max 1 2 1 Exp U U U n Y L n n = . P.166
Stat 1301B Probability & Statistics I Fall 2011-2012 Converges in probability n X is said to converge in probability to if for any X 0 > ε , ( ) 0 lim = X X P n n . It is denoted as . X X P n ⎯→ Example 4.2 Consider the defined in example 4.1. Obviously, n X ( ) 0 1 = n X P if 1 > . For any 1 0 < , () ( ) ( )( ) n X n n n n F X P X P X P εε = = = = 1 1 1 1 1. Therefore for any 0 > , ( ) 0 1 lim = n n X P and hence . 1 P n X Converges Almost Surely / Strong Convergence n X is said to converges almost surely to X if ( ) 1 lim = = X X P n n It is denoted as . X X s a n . .

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Chapter4 - Stat 1301B Probability Statistics I Chapter IV...

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