functions of random variables

Ngày tải lên: 17/12/2013, 15:18

... 10 Completion of the proof of sufficiency 240 11 Proof of the necessity 241 12 Completion of the proof 243 Chapter 13 Monomial zones of integral attraction to Cramer's system of limiting tails ... 171 Statement of the theorem The introduction of auxiliary random variables 172 Proof of the theorem 174 Chapter Monomial zones of local normal attraction 177 177 178 180 Zones of normal attraction ... the Borel sets of R" is the distribution of X, or the joint distribution of the variables X1 , X2 , , X,, More generally, if T is any set of real numbers, a family of random variables X (t),...

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 ... < ε0 < in theorem 1.1 of [12] Remark 1.4 If EX < ∞, then X 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 ... sequence remains unknown The purpose of this article is to study and establish the ASCLT for self-normalized partial sums of random variables in the domain of attraction of the normal law, we will show...

Ngày tải lên: 20/06/2014, 21:20

... Definition 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 , ... ξ· inf ξ 1.2 Definition 1.3 If ξ1 , , ξn are bounded random variables, the independent arithmetic mean of the product of random variables ξ1 , , ξn , A ξ1 , , ξn is given by n i n sup ... ξn n A ξi , i G ξ1 , , ξn n G 1.5 ξi i The mean 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...

Ngày tải lên: 22/06/2014, 02:20

... fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions fi(t), then ... distributions can arise as limits of distributions of sums of independent random variables Consider, for each n, a collection of independent random variables, Xnl , Xn2, , Xnkn The Xnk are said ... class of complex functions A (x) with bounded variation Proof 00 I dA (x) I < co , V (A) = Var A = J and by V the class of Fourier-Stieltjes transforms 00 e`tx dA (x) a (t) = ~ - 00 of functions...

Ngày tải lên: 27/06/2014, 03:20

... fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions fi(t), then ... distributions can arise as limits of distributions of sums of independent random variables Consider, for each n, a collection of independent random variables, Xnl , Xn2, , Xnkn The Xnk are said ... class of complex functions A (x) with bounded variation Proof 00 I dA (x) I < co , V (A) = Var A = J and by V the class of Fourier-Stieltjes transforms 00 e`tx dA (x) a (t) = ~ - 00 of functions...

Ngày tải lên: 02/07/2014, 20:20

... 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 F (a x + b 2), while that of the right-hand ... X2+ + Xn of independent random variables with distribution F has distribution function of the form F (a n x + b n), so that X1 + +Xn _ an On has distribution function F The proof of the final ... indicate the rather simple form of the characteristic functions of stable laws The bulk of the chapter is devoted to the investigation of the analytical properties of the corresponding densities,...

Ngày tải lên: 02/07/2014, 20:20

... concentrated on a set of zero Lebesgue measure) with S' (x) = for almost all x Then the density of x is p (x) = F' (x) = ap (x) Now let X , X2 , be a sequence of independent random variables with ... in variation § A refinement of the local limit theorems for the case of normal convergence In this section we assume that the common distribution of the random variables X; has zero mean and ... the common distribution function F(x) of the Xj should belong to the domain of attraction of the stable law, and (2) there exists N with sup p N (x) < oo Proof Condition (2) is clearly necessary,...

Ngày tải lên: 02/07/2014, 20:20

... then IIFn-Gall -* o This chapter is devoted to a study of the analogous problem in the space DOMAINS OF ATTRACTION OF STABLE LAWS LP of functions Ilfii,= f J f 141 for which the norm If(x)Ipdx ... of attraction of G (of exponent a) such that, for all n and all p < a -1, 00 IF„(x)-G(x)IPdx = oo 00 Suppose therefore that F is in the domain of attraction of a stable law G - of exponent a, ... terminology of § we could define the L P domain of attraction of G as the aggregate of distributions F with IIF n -GII P-+0 The theorem shows that, so long as p > a -1 , the LP domain of attraction...

Ngày tải lên: 02/07/2014, 20:20

... we can speak of the "domain of attraction" of the "limiting tails" The problem of discovering the possible forms of the limiting tails is closely analogous to the classical problem of characterising ... hold when x i 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 ... normalised sums of independent variables, i e the stable laws And of course there is a corresponding problem of local limit theorems In the following chapters, several systems of limiting tails...

Ngày tải lên: 02/07/2014, 20:20

... limit theorem for lattice variables We now proceed to the proof of Theorem 7.1 We introduce 00 M (z) = E (ezX i) = P k exp [(kh + b) z] , k=-co defined in IRe z i < a because of (7.1 2), and periodic ... coefficients of this series determine the first (k+3) moments of Xj (assuming that EXj =O and that 62 = VXj is known) In fact if these coefficients are known, we have the first (k + 3) terms of the ... THEOREMS ; BERNSTEIN'S INEQUALITY 162 Chap lemma from the theory of Fourier transforms (for the proof of which see [11], page 20) Lemma 7.2 If a bounded continuous function g (x) E L (- oo, oo) has...

Ngày tải lên: 02/07/2014, 20:20

... „ 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

... The case of (9.2 3) is treated similarly Theorem 2 For random variables of class (d) the condition (9.2 1) is necessary in order that [0, n" p (n)] and [- n" p (n), 0] should be zones of local ... the moments of X X, up to order (s+3), should coincide with those of a normal distribution Conversely, these two conditions suffice for [ - n"/p (n), 0] and [0, n"/p (n)] to be zones of local normal ... zones of local normal attraction THE FUNDAMENTAL THEOREMS 181 Proof In view of Theorem 2.2, it is sufficient to consider variables satisfying (9 2.1), but we shall use only the weaker assumption...

Ngày tải lên: 02/07/2014, 20:20

... and (10.1 11) follow) These theorems are, of course, of collective type in the sense of Chapter ; the moments of X~ (up to order s+ 3) play the role of the linear functionals aj, bi We see that, ... every sequence of numbers can be the sequence of coefficients of a Cramer series „ On the condition (10 9) We first show that (10.1 9) follows from (10 1.12) ; the proof follows that of Theorem ... „ Derivation of the fundamental integral We now proceed to the proof of Theorem 10 1, assuming as we may that o =1 and replacing (10 9) by the weaker condition 10 DERIVATION OF THE FUNDAMENTAL...

Ngày tải lên: 02/07/2014, 20:20

... Thus the sum of (11 9.9) over < r < m is Bexp(-4n 1-2 "`)= B exp{-c exp (2XP (n))} (11 9.10) 11 10 COMPLETION OF THE PROOF OF THEOREM 11 21 A similar analysis can be made of n 1/2 -° f- ... completes the proof of Theorem 11 2.2 in the special case of monomial zones We now proceed to the general case „ 15 The general case of narrow zones We now follow the argument of „„ 11-14 to ... is a zone of normal attraction This completes the proof of Theorem 11 2.2 „ 16 The transition to Theorems 11 2.3-5 The remaining theorems refer to the "very narrow" zones Functions of Class...

Ngày tải lên: 02/07/2014, 20:20

... COMPLETION OF THE PROOF 243 „ 12 Completion of the proof Now suppose that [0, no'p (n)] and [ - n" p (n), 0] are zones of normal attraction, and consider the distribution function F§ (x) of n (S§ ... required integral theorem „ 11 Proof of the necessity We now complete the proof of Theorem 12 by showing that, if [0, n" p (n) ] and [ no 'p (n), 0] are zones of normal attraction, then (12 2.1) ... Theorem 12.1 can be proved by the methods of Chapter „ An upper bound for the probability of a large deviation We now proceed to the proof of the sufficiency part of Theorem 12 2, assuming (12 3) and...

Ngày tải lên: 02/07/2014, 20:20

... 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 ... )± K3 is a truncated Cramer series By virtue of the definition of z', LK](h)-h dd LK J(h)= - 22 +z ) "(z)+Bn -3 (13 14) COMPLETION OF THE PROOF 253 Substituting this shows that (13 3.11) ... Completion of the proof Inserting (13 15) into (13 10), we have 00 1-Fn(x)=exp(-2ni2+ni3At~K](i)+Bn-2) e-n•"1/z•dFn(v)+ f0 + ep n3 (13 4.1) For the computation of the integral we follow the argument of...

Ngày tải lên: 02/07/2014, 20:20

... coefficients of the rational function are expressed in terms of a finite number of the derivatives at zero of the radial extension y (t) of 0(t) These derivatives are called the pseudomoments of X, ... k=r +0(~ -K-1 ) (14.3 7) l+ The question of the differentiable of the radial extensions of (14 5) therefore reduces to that of the radial extensions of 00 e it4 ~k (14.3 8) -00 ( +Ok 60 INTEGRAL ... values of y for which the probability that two or more of the events (14 2.2) occur is of order Bi7nny -a , (14 2.3) where i , -*0 as n-> oo For each k >, the probability that exactly k of the...

Ngày tải lên: 02/07/2014, 20:20

... Is'- (15.2.1) 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 ... 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 ... = then Q (L) < C L/l0 Proof The concentration functions of S§ and S§ - E 1'i coincide Proof of Theorem 15.2.1 First suppose that the distribution functions Fi (x) of the Xi are continuous and...

Ngày tải lên: 02/07/2014, 20:20

... non-negative values of t are considered, a semigroup of isometric operators) In the discrete time case, this is the cyclic group of powers of the restriction U of Ti In fact, because of (16 2.3) and ... We state two theorems on the representation of groups of unitary operators as Fourier-Stieltjes transforms of partitions of the identity (the proof of which can be found in [2]) Theorem 16.3 ... limit of a sequence (fn) of step functions Because I is an isometry and L2 (Q) is complete, the elements I (fn) will converge to some element of L2 (Q) This element is independent of the choice of...

Ngày tải lên: 02/07/2014, 20:20