By Malempati M. Rao, Randall J. Swift
This is a revised and increased version of a winning graduate and reference textual content. The ebook is designed for the standard graduate path on likelihood concept, together with a few very important purposes. the hot variation bargains a close therapy of the middle region of likelihood, and either structural and restrict effects are awarded intimately. in comparison to the 1st variation, the fabric and presentation are larger highlighted; each one bankruptcy is greater and updated.
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This becomes > + 4(yo) g(yo)(y yo), the right side is In this inequality, written as 4(y) called the support line of 4 at y = yo. Let X(w) = y, and yo = E ( X ) in (13). , and taking expectations, we get This is (12), and the result holds. [Note: t i < tz - +-g(t1) < g ( t ~ ) . ~ ] This is not entirely trivial. Use ( 3 ) in different forms carefully. H. E. Littlewood, and G. PolyA (1934, p. ] 1 Background Material and Preliminaries 18 In establishing (10) we first showed that &/' = [ E ( I X I ~ )is] an ~ Iincreas~ ing function of r.
If X I , X 2 , . . is a sequence of random variables, then o ( X i , i n ) is called the tail a-algebra of {X,, n 1). n,"==, > > Theorem 7 (Kolmogorov's Zero-One Law) A n y event belonging t o the tail a-algebra of a sequence of independent random variables o n ( 0 ,C , P) has probability either zero o r one. > Proof Denote by 7 = a ( X k ,k n), the tail a-algebra of the sen 1) are independent quence. Then by Theorem 3a, a(X,) and a ( X k ,k a-algebras for each n 1. But 7 c a ( X k ,k n I ) , so that a(X,) and 7 are independent for each n.
We then have T h e o r e m 4 (Egorov). Let {X,X,,n > 1) be a sequence of random variables o n ( R , C, P). e. iff the sequence converges t o X P-uniformly. Proof One direction is simple. In fact, if X, + X P-uniformly, then for l / n O there is an A,, E C with P(A,,) < l / n O and X,(w) + X(w) A,, theii P ( A ) = 0, aiid if w E fl A, uniformly on R A,,. e. The other direction is non-trivial. e. Then there is an N E C, P ( N ) = 0, and X,(w) + X(w) for each w E R - N . If k > 1, m > 1 are integers and we define E = n,,, - Ak,, = {w E 0 - N : IX,(w) - X ( w ) - 1 m < - for all n > k), then the facts that X, + X on 0 - N and Ak,, c &+I,, imply that R-N = U =,: Ak,, for all m 1.