{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

3advancedProb

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

{[ snackBarMessage ]}

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon bookmark
Ask a homework question - tutors are online