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1 Stat 643 Review of Probability Results (Cressie) Probability Space: ( , , ) H T T is the set of outcomes H is a -algebra; subsets of T 5 H is a probability measure mapping from onto [0,1]. T T Measurable Space: ( , ). H T Random Variable: Su ppose ( , , ) is a pr obabil it y sp ace and le t : be meas ur able (i .e., H T H ‘ T \ Ä { : ( ) } ). Then is said to be a random variable (r.v.). = H = ! T - \ Ÿ - \  Integral of a Measurable Function: Suppos e ( , , ) is a meas ur e sp ace (i .e., maps fr om onto H T . . T [0, ]) and is a measur able mappin g. Then is defined as a limi t of integr als of simpl e (i.e. , _ 0 0 . ' . step) functions: Write: , where , 0. If and then the integral is said to be 0 œ 0 • 0 0 0 0 . _ 0 . _ + + + • • • ' ' . .  finite not to exist  . If , then the integral is said ; otherwise it is said to ' ' 0 . œ _ œ 0 . + . . • exist. The measurable function is said to be if . 0 0 . integrable exists and is finite ' .  Notation: If is a r.v. on ( , , ), write E( ) for . \ T \ \.T  H T ' Important Convergence Theorems Le t ( , , ) be a measure space an d ( , ) be the me as ur ab le space of rea l nu mb ers wit h th e Bo re l - H T . ‘ U 5 1 algebra . In the following, , , { } , and { } denote measurable functions U 1 1 1 1 0 0 1 8 8 8 8 > > from ( , ) into ( , ). H T ‘ U  1 Fatou's Lemma: If 0 a.s. ( ), for all 1, then 0 8 8 . ' ' liminf liminf . 8 Ä _ 8 Ä _ 0 . Ÿ 0 . 8 8 . . Monotone Convergence Theorem: Suppose that a.s. ( ), 0 . Then . Ÿ 0 Å 0  8 ' ' 0 . Å 0 . 8 . . . Dominated Convergence Theorem: Suppose that a.s. ( ), as and | | for all . 0 Ä 0 8 Ä _ 0 Ÿ 1 8 8 8 1. _ 0 0 . œ 0 . 8 Ä _ 1. If , then is integrable and lim . ' ' ' . . . 8 Extended Dominated Convergence Theorem: Suppose that ) a.s. ( ), a.s. ( ). (i 0 Ä 0 1 Ä 1 8 8 . . ) | | a.s. ( ) and , for all 1. (ii 0 Ÿ 1 1 . _ 8 8 8 8 . . ' ) lim . (iii 8 Ä _ 1 . œ 1. _ ' ' 8 . .

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Page 1: Condition A Ex of Convergence Theorem

8/6/2019 Condition A Ex of Convergence Theorem

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