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Unformatted text preview: Key Concepts in Advanced Probability Peter Glynn ([email protected]) 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 (Minkowski’s inequality), and X 1 X 2 1 ≤ X 1 p X 2 q for 1 /p + 1 /q = 1 with p, q ≥ 1 (H¨older’s 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 Markov’s 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 Chebyshev’s inequality. If E [exp( θW )] < ∞ , put X = exp( θW ); this yields the inequality P { W > w } ≤ exp(- θw ) E [exp( θW )] . Hence, we obtain the exponential inequality P { W > w } ≤ inf θ ≥ exp(- θw ) E [exp( θW )] . 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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