3advancedProb

3advancedProb - Key Concepts in Advanced Probability Peter...

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Unformatted text preview: Key Concepts in Advanced Probability Peter Glynn (glynn@stanford.edu) April 8, 2007 1 Expectations The expectation E [ X ] of any non-negative random variable X can always be defined uniquely; the expecta- tion may be either finite or infinite. If X is of arbitrary sign, put X + = max( X, 0) X- = max(- X, 0) E [ X ] is said to exist if at least one of E [ X + ] and E [ X- ] are finite, in which case we put E [ X ] , E [ X + ]- E [ X- ]. If E [ | X | ] < then X is said to be integrable . If X = g ( Y ) (for g : IR IR and Y a real-valued r.v.), then E [ X ] can be computed as a Stieljes integral: E [ X ] = Z IR g ( y ) F ( dy ) , where F is the distribution function of the r.v. Y given by F ( y ) = P { Y y } . Note that if Y has a density f , then E [ X ] = Z - g ( y ) f ( y ) dy, whereas if Y has a probability mass function p then E [ X ] = X y g ( y ) p ( y ) . 2 Useful Inequalities For p 1, put X p , E [ | X | p ] 1 /p . Then, we have X 1 + . . . + X n p X 1 p + . . . + X n p (Minkowskis inequality), and X 1 X 2 1 X 1 p X 2 q for 1 /p + 1 /q = 1 with p, q 1 (Holders inequality). The special case with p = q = 2, namely X 1 X 2 1 X 1 2 X 2 2 , is called the Cauchy-Schwarz inequality. 1 If X is nonnegative, P { X > x } x- 1 E [ X ] for x > 0; this is called Markovs inequality. If E W 2 / < , then we can set X = ( W- E [ W ]) 2 to yield P {| W- E [ W ] | > w } var ( W ) /w 2 ; this special case is called Chebyshevs inequality. If E [exp( W )] < , put X = exp( W ); this yields the inequality P { W > w } exp(- w ) E [exp( W )] . Hence, we obtain the exponential inequality P { W > w } inf exp(- w ) E [exp( W )] . 3 Weak Convergence Let ( X n : 1 n ) be a sequence of finite real-valued random variables. Then, X n converges weakly to X (also known as convergence in distribution ) if P { X n x } P { X x } as n at each x at which P { X } is continuous. We use the notation X n = X (or, equivalently, P n = P where P n ( ) = P { X n } ) to denote weak convergence. Weak convergence can be re-formalized in several equivalent ways....
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This note was uploaded on 08/06/2008 for the course CME 308 taught by Professor Peterglynn during the Spring '08 term at Stanford.

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3advancedProb - Key Concepts in Advanced Probability Peter...

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