Limit_Theorems

Limit_Theorems - Limit Theorems We begin by reviewing the...

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Unformatted text preview: Limit Theorems We begin by reviewing the result that says that the correlation between two random variables , X Y is always between -1 and 1. This derives from the so-called Cauchy- Schwartz Inequality (also know as the Cauchy-Bunykowski-Schwartz inequality or, simply, Schwartzs inequality). Theorem 1 : Let X and Y have bivariate density ( 29 , f x y , and suppose that E XY exists. Then 2 2 E XY E X E Y Proof : Since the random variable 2 2 2 X Y E X E Y - is nonnegative, we have ( 29 ( 29 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 X Y E E X E Y XY X Y E E X E Y E X E Y E X E Y E XY E X E Y E X E Y E XY E X E Y - = +- = +- =- Hence, ( 29 2 2 1 E XY E X E Y , and the result follows. In terms of covariance, 1 ( 29 [ ] ( 29 [ ] ( 29 ( 29 ( 29 [ ] ( 29 ( 29 [ ] ( 29 [ ] ( 29 2 2 , X Y Cov X Y E X E X Y E Y E X E X Y E Y E X E X E Y E y =-- -- -- so ( 29 ( 29 , , 1 X Y Cov X Y X Y = Recall also that, since ( 29 [ ] ( 29 2 2 Var X E X E X =- , it follows that for any two random variables, X and Y , ( 29 ( 29 2 E X Y E X Y - - . Closeness of Random Variables Recall that a random variable is a real-valued function whose domain is a sample space S . There are many ways to measure the distance between two functions. As we consider these, you should think in terms of the graphs of the functions. We start with a simple example. Example 1: Let ( 29 m g x x = and ( 29 n h x x = for 0 1 x . In this example, both g and h are continuous, so ( 29 ( 29 { } max :0 1 g x h x x- exists. This is simply the largest vertical distance between the graph of g and the graph of h . It is called the L -distance between g and h , and is denoted by g h - . The graph that follows shows this distance for the functions ( 29 g x x = and ( 29 3 h x x = . The maximum distance occurs for 6 2 3 x = , and 0.1481 g h - = . Notice that this distance is completely determined by a single value of x , and does not take the whole graphs of g and h into account....
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This note was uploaded on 04/09/2009 for the course MATH 40051 taught by Professor Mocioalca,o during the Fall '08 term at Kent State.

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Limit_Theorems - Limit Theorems We begin by reviewing the...

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