b 8 functional transformations of random variables

... F, is the distribution of the normalised sum (X1 + + X©-mn)/Qn- , (8. 3 8) PROOF OF THE THEOREM 175 so that we can use the theorems of „ 3.5 From (8. 3 3), a + O (h) , = (8. 3 9) and Fn (y) ... REFINEMENT BY PETROV Chap 172 „ The introduction of auxiliary random variables Since E (exp a JXX j) < oo , we may write, for Jhj

Ngày tải lên: 02/07/2014, 20:20

... n-+oo, BM Bn a2 al m Bn 2 CANONICAL REPRESENTATION OF STABLE LAWS 39 Consider the sum Bn X1 + +Xn B Bn - A" b1 + Bn B Xn+i+ +Xn+m BM Xl + + Xn+m - A , (2 1.6) B where B = Bn/a , A = (Bn ... independent random variables § Probability spaces, conditional probabilities and expectations A probability space is a triple (Q, R, P), where Q is a set of elements w, R a a-algebra of subsets of Q ... a-algebra of Borel sets, P the Lebesgue-Stieltjes measure determined by P { [a, b) } = F (b) - F (a), and X (w) = w 21 CONVERGENCE OF DISTRIBUTIONS Let X and Y be independent random variables...

Ngày tải lên: 08/04/2014, 12:28

... sums of random variables in the domain of attraction of the normal law Qunying Wu1,2 College of Science, Guilin University of Technology, Guilin 541004, P R China Guangxi Key Laboratory of Spatial ... wqy666@glite.edu.cn Abstract Let X, X1 , X2 , be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution A universal result ... is in the domain of attraction of the normal law Therefore, the class of random variables in Theorems 1.1 is of very broad range Remark 1.5 Essentially, the open problem should be whether Theorem...

Ngày tải lên: 20/06/2014, 21:20

... inequality of Polya-Szeg¨ : o m k where ak , bk > 0, k a2 k m k 1, , m, a bk ≤ ak , A AB ab ab AB max ak , b m ak bk , 3.7 k bk , and B max bk Mingjin Wang Theorem 3.4 the extensions of Kantorovich’s ... inequality of two random variables Theorem 1.6 Let ξ and η be bounded random variables If inf ξ > and inf η > 0, then Eξ ·Eη2 A ξ, η ≤ E2 ξη G ξ, η 1.6 Equality holds if and only if P ξ η a B ξ η ... 1.4 If ξ1 , , ξn are bounded random variables with inf ξi ≥ 0, i 1, , n, one defines the independent geometric mean of the product of random variables ξ1 , , ξn to be n G ξ1 , , ξn sup...

Ngày tải lên: 22/06/2014, 02:20

... n-+oo, BM Bn a2 al m Bn 2 CANONICAL REPRESENTATION OF STABLE LAWS 39 Consider the sum Bn B X1 + +Xn Bn - A" b1 + Bn B Xn+i+ +Xn+m BM Xl + + Xn+m B where B = Bn/a , - A , (2 1.6) A = (Bn ... a-algebra of Borel sets, P the Lebesgue-Stieltjes measure determined by P { [a, b) } = F (b) - F (a), and X (w) = w 21 CONVERGENCE OF DISTRIBUTIONS Let X and Y be independent random variables ... deduce that, as n-+ co, lim BB n t 14) (t) = lim Bn n+1 t)10(t) I = (2.1 5) If Bn+1 /Bn +l,we can find a subsequence of either (Bn+1 /Bn) or (Bn /B n+1 ) converging to some B < Going to the limit...

Ngày tải lên: 27/06/2014, 03:20

... a-algebra of Borel sets, P the Lebesgue-Stieltjes measure determined by P { [a, b) } = F (b) - F (a), and X (w) = w 21 CONVERGENCE OF DISTRIBUTIONS Let X and Y be independent random variables ... probability one, E{E(X IUI R1} = E{X IR1} § Distributions and distribution functions If X is a random variable, its probability distribution is the measure F (A) = P (X E A) on the Borel subsets ... distributions of sums of independent random variables Consider, for each n, a collection of independent random variables, Xnl , Xn2, , Xnkn The Xnk are said to be uniformly asymptotically negligible...

Ngày tải lên: 02/07/2014, 20:20

... n-+oo, BM Bn a2 al m Bn 2 CANONICAL REPRESENTATION OF STABLE LAWS 39 Consider the sum Bn X1 + +Xn B Bn - A" b1 + Bn B Xn+i+ +Xn+m BM Xl + + Xn+m - A , (2 1.6) B where B = Bn/a , A = (Bn ... EBn Bn 00 x dx (x) S0 =1 Moreover, 00 dx(z) nx(xBn) = n xB„ 2s+I x B 00 Y 2-2s x -2Bn °° E -2n2 nxBH (xBn) z dx(x) C 2sxB„ S=0 ( 2s+ xB -2s H ) H(2sxB n) - H + xBn) (2s " H(xBn) s=o (2 6. 18) ... (Bn A n + Bm A m + b Bn + b Bm) /B From the assumption of the theorem, the distribution functions of the two components of the left-hand side of (2 6) converge respectively to F (a l x + b 1) and...

Ngày tải lên: 02/07/2014, 20:20

... Bnx+AnBn-ANBNI BN *m * * F *r (Bn X +An Bn) = ajbm-jH*j B n x+A"Bn - ANBN BN j=0 (J) M m * (Bnx+AnBn-ANBN * *HZ (m_j) F *r (Bn x+A n Bn) _ BN _ j-ma Ym l/2logm m + j-ma> -m'/2logm ajbm-j Hlj* ... proof of the theorem Any integer n > N can be written n = mN + r, where m and r are integers and < r < N By Lemma 4.4.2, F,,(x) = F*" (B n x + A n B n) = Bnx+AnBn-ANBN BN + bH2 {aHl Bnx+AnBn-ANBNI ... ItI 2ma t BN bB n ( Bn BN ItI _ ajb'n 'dt j-ma> -m /2logm l Bn IhI m Y h (t)I 2ma J dt (4.4.15) > SBN Because h l (t) is the characteristic function of an absolutely continuous distribution, there...

Ngày tải lên: 02/07/2014, 20:20

... attraction of stable laws in the L metric P Let X1 , X2 , be a sequence of independent random variables with the same distribution F If it is possible to select normalising constants A,,, B in ... G) belongs to LP Lemma 2.2 If F belongs to the domain of attraction of a stable law G of exponent a, then for all < a the moments 00 Kn(a) = f- IxI a dFn(x) 00 are uniformly bounded in n Proof ... we denote by L,,,, the Banach space of bounded measurable functions f on (- oo, oo), with norm 11111= ess sup I f (x) I , then (5 1) asserts that, when F is in the domain of attraction of Ga ,...

Ngày tải lên: 02/07/2014, 20:20

... estimation of the probabilities of large deviations (cf [ 18] , [ 185 ]) Sometimes it is necessary to give bounds for such probabilities in wider ranges x=0(ni), in which the case of the Bernoulli ... n)-+1 , (6 2) P(Z n < - x)/O(-x, b , b , , bl, n)-+1 , (6.2.3) where the parameters a , , ak , b1 , , b, are linear functions of the distribution F of the variables X; Such a limit theorem ... is of order n2 ; the correct asymptotic expression includes terms of entropy type „ Statement of the problem For the variables X; introduced at the beginning of the chapter, we examine the behaviour...

Ngày tải lên: 02/07/2014, 20:20

... 2(T)}(1+Bn -1 log8 n) (7 3.5) Furthermore, (7.2.7) gives K" (zo) = U2 + Bzo = 0- + BT ( 7.3 6) 7.4 A LOCAL LIMIT THEOREM FOR LATTICE VARIABLES 67 Substituting into (7.3 5) and noting that (1+Br)(1+Bn ... {1-ri(c)}"-2 - ~r~ >E ~M(it)1 dt o0 B{ 1-1i(c)} n-2 (7.2.3) Here B is bounded and r1 (c) > The right-hand side of (7.2.3) can be written as B exp (- nri (c)), where ri l (c) > Substituting into (7 2.2) ... (27r/nK"(z o))2 (1 +Bn -1 log8 n) Thus the first term on the right-hand side of (7.2.25) is equal to a( 2itK"(zo)) -I- exp In (K(zo) - aTzo)}(1+Bn -1 log8 n) , (7.3 3) (7 4) or, because of (7.2.19),...

Ngày tải lên: 02/07/2014, 20:20

... (1-2p)] = =B exp [q (B+ 1-2p){2-pl) log n+log C1 +p log q] Multiplying this by n- -!q = exp (-2q log n) , and by Xq q! (9 6.5) (9 6.6) = B exp (pl q log n) , we obtain the expression B exp[q (B+ (1-2p)(2-p ... exp(-in tx)dt+R , (9 7) MONOMIAL ZONES OF LOCAL NORMAL ATTRACTION 182 Chap where JR, I =B e" (9 8) Because of (9.3 2) 0(t) is infinitely differentiable for all t For positive integers T, p ... 1) m log m - ° m log n) _ = B exp (Bm+(k-1) m log m-(2-a ) m log n) _ Bm+m =B exp -2a (2a log n-log Ka ) log n = 4a = B exp m B- (a-a ) log n - 1-2a log K 4a (9 4.17) But a >, a, 1- 2a > 0, so...

Ngày tải lên: 02/07/2014, 20:20

... in"p(n)) < exp [-8ao n (10.2 1) The event whose probability is so bounded certainly occurs if both of the independent events IX2 + +XJ/Gn2

Ngày tải lên: 02/07/2014, 20:20

... 18 3) Now BrF(2r)n - « - ul ) r= B exp [r (B+ 2 log r-(e-u ) log n)] , ( 11 18 4) log r-(e-° 1) log n

Ngày tải lên: 02/07/2014, 20:20

... Let X1 be independent random variables with probability density d17(x) = R -l e (hx)p(x) , dx (12.3.5) where R = For n l _ e (hy) p(dy) (12.3.6) n, write f© (x) for the probability density of S;, ... will be true, for example, if F (x) = P (XX < x) contains an absolutely continuous-component, in which case Theorem 12.1 can be proved by the methods of Chapter „ An upper bound for the probability ... (and ; like B, may vary from place to place), and p n1 is some error bound, attempting to establish these by induction on n1 12 231 STUDY OF THE BASIC RELATION „ Study of the basic relation...

Ngày tải lên: 02/07/2014, 20:20

... n2 x (1+Bn -2)(1+Bh)+6p n3 , (13.4.3) or, because of (13 3.9) and (12 10.5), 1- Fn (x) = { 1- (x) } exp x~ ~= n2 K~ x n2 x x (1+Bn-2)(1+Bh)(1+Bx-1) (13 4.4) But x > n2 and h = Bn- x = Bn"- 2/ ... formulae of the type (13.1 2), (13 3) Further, in the collective Theorem 13 1 the role of the linear functionals a j , is played by the moments of Xj „ An upper bound for the probability of a large ... also In view of the definition of K, we can replace [2K] by (s], to obtain (13 2) Finally (13 3) is derived by replacing X by -Xi The theorems of this chapter of course contain those of Chapter...

Ngày tải lên: 02/07/2014, 20:20

... is b times differentiable, (q) y(t)= 1-2t + Y Y 0) b - q =3 in JtI e o If l tl nBt b tq+Bt (14.4.6) b q ,< n - log n, = nB(n -ib log n )b = BEn- 26+ i +E , ( 14.4.7) and since exp (B - ,b+ E 1 +B ... class of such probability densities we call (A) Such variables have only a finite number of moments, and the role of the linear functionals aj , b i is layed by pseudomoments defined in „ below ... function in both arguments For x>,nZ+a-1+E, n>n o (8) , (2rc) -Z fx e -Za2 du+r(x,ni)-nP(X j >axn+) ; (14.1 8) r(x, n+) is determined by a finite number of linear functionals of the distribution of X1...

Ngày tải lên: 02/07/2014, 20:20

... the members of K o each have size m Thus the number of members of Ko cannot exceed the number of subsets of 2C of size ... The proof of this theorem requires a number of auxiliary results Lemma 15.2.1 Let 21 be a set of n elements, and K a class of subsets of 21 that no member of K is contained in any other member ... given set more than once ; enumerate the distinct members of the collection as B , B2 , , BS Each B a can be a subset of at most (n - k + 1) of the A , and so can appear at most (n - k + 1) times...

Ngày tải lên: 02/07/2014, 20:20

... with probability 1, then Ti (fi n )-+Ti (~) with probability Since each random variable ~, measurable with respect to M, is the limit as n ' co with probability 1, of the random variables _ j ... class of random variables measurable with respect to fit, defined by the conditions (1 ) Ti is well-defined up to differences on sets of zero probability (2 ) If X (A) is the indicator function of ... STRUCTURE OF L,p AND LINEAR TRANSFORMATIONS 297 Theorem 16 6.1 The space L,,,, consists exactly of the random variables of the form (16.6.1) In fact, we can say more than this Because of (16 5 .8) ...

Ngày tải lên: 02/07/2014, 20:20

... terms by Y_', and over negative terms by E - , we have ~IP(BiIA j)-P (B) I= i {P(Bil Aj)-P (B, )} Bi Aj)-PC _~ P i U+ i U+ Bi {P(BiI Aj) - P(Bj)} _ l }+ i JBi Aj - v - Bi 20 (t) (17.2.14) i Substituting ... supremum of a random variable ~ (w) is the unique number C < oc with the property that P ( > C) = but P (> C') > for all C'< C We remark that, if sup ess sup IP(BI9Jtr j -P (B) I , Cpl (T) = (17 2.9) BESUtt+ ... P(BE)I > 01(r) - E IP(BEI` _ Without loss of generality we may suppose that, for all w e AE , P(BE I9JI`-~)-P (B E) > (r)- E Integrating this inequality over A, we obtain P (AE BE) - P (AE BE)...

Ngày tải lên: 02/07/2014, 20:20