basic concepts of set theory
A Course in Mathematical Statistics
... other sets in question will be subsets of S 1.1.1 Set Operations The complement (with respect to S) of the set A, denoted by Ac, is defined by Ac = {s ∈ S; s ∉ A} (See Fig 1.2.) Basic Concepts of Set ... Chapter Basic Concepts of Set Theory 1.1 Some Definitions and Notation A set S is a (well defined) collection of distinct objects which we denote by s The fact that s is a member of S, an element of ... Concepts of Set Theory 1.2* Fields and σ-Fields In this section, we introduce the concepts of a field and of a σ-field, present a number of examples, and derive some basic results DEFINITION A class (set) ...
- 593
- 949
- 0
independent and stationary sequences of random variables
... 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 ... 14 Completion of the proof of Theorem 11 2.2217 15 The general case of narrow zones 218 16 The transition to Theorems 11 2.3-5220 17 Choice of p 222 18 Completion of the proof 224 Chapter ... defined on 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...
- 438
- 855
- 0
Báo cáo toán học: " A note on the almost sure limit theorem for self-normalized partial sums of random variables in the domain of attraction of the normal law" pptx
... 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 ... in a more general setting Theorem 1.1 Let {X, Xn }n∈N be a sequence of i.i.d random variables in the domain of attraction of the normal law with mean zero Suppose ≤ α < 1/2 and set dk = exp(lnα ... < 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...
- 13
- 480
- 0
Báo cáo hóa học: "Research Article Recurring Mean Inequality of Random Variables" doc
... ξk , 2.8 Journal of Inequalities and Applications Some applications In this section, we exhibit some of the applications of the inequalities just obtained We make use of the following known ... lemma which we state here without proof Lemma 3.1 If < m2 ≤ m1 ≤ M1 ≤ M2 , then 1/2 m1 M1 m1 M1 M2 1/2 m2 ≤ m2 M2 3.1 Theorem 3.2 the extensions of the inequality of Polya-Szeg¨ Let aij > 0, o maxj ... References M Shaked and Y L Tong, “Inequalities for probability contents of convex sets via geometric average,” Journal of Multivariate Analysis, vol 24, no 2, pp 330–340, 1988 M Shaked and J...
- 6
- 112
- 0
Independent And Stationary Sequences Of Random Variables ppt
... characteristic function f of F For the proof of this theorem, see for example [47] In the sequel we shall need various refinements of this theorem permitting us, from the proximity of their characteristic ... 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 ... 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,...
- 426
- 321
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 1 pptx
... characteristic function f of F For the proof of this theorem, see for example [47] In the sequel we shall need various refinements of this theorem permitting us, from the proximity of their characteristic ... generalisation of that of an integer-valued random variable The maximal value of h for which the distribution is concentrated on an arithmetic progression with step h is called the step of the distribution ... over all Borel sets A A sequence of distributions F, converges in variation to a distribution F if p (F", F)-*0 It is clear that this mode of convergence can be expressed in terms of distribution...
- 20
- 374
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 2 ppt
... 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 ... 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, ... method of proof is that of contour integration and, later, the technique of steepest descent Because of (2.3 2) we can, and consistently will, restrict attention to positive values of x Theorem...
- 57
- 299
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 4 ppt
... sufficient that (1) the common distribution function F of the X X should belong to the domain of attraction of G, and (2) the interval h be maximal Proof The transformation X,'= (X;-a)/h permits us ... 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, ... distribution 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...
- 19
- 345
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 5 pot
... 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 ... distribution G is a limit in LP (p > 0) of distributions of normalised sums of the form (5.2.1), then it is necessarily stable Conversely, if the distributions F,, of (5 2.1) converge weakly to a stable...
- 15
- 310
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 6 pptx
... 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 ... behaviour of the tails of the distribution of Z", in the range lxi < n" (a < 2), is determined for distributions satisfying Cra- INTRODUCTION AND EXAMPLES 157 mer's condition by a finite number of ... an upper bound of wide applicability Let us remark that the study of the very large deviations x = (n ) gives rise to an expression involving the entropy of a certain system of events (Sanov...
- 6
- 323
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 7 potx
... function of X - X2 , which has a bounded continuous density g (x) Then (7 2.1) follows from the following RICHTER'S LOCAL THEOREMS ; BERNSTEIN'S INEQUALITY 162 Chap lemma from the theory of Fourier ... 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 ... 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 with period 27ri/h...
- 11
- 319
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 8 pps
... y) dF© (y) Now 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 ... ; ITS REFINEMENT BY PETROV 174 Chap „ Proof of the theorem Since log R = log ~~ 00 e"''dV (y) _ y Y'' by , y12 v! y (8 1) where y are the cumulants of the X (7 2.6), we have d Y, i Fn = - log ... (8.3.3) ( In the notation of „ 2, logR=K(h), so that the factor multiplying the integral (8 2.12) is exp {n (K (h) - hK' (h)) } ( 8.3 4) We now choose h to be the solution of the saddle point equation...
- 6
- 336
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 9 ppsx
... Derivation of the basic integral Write cl r = K ( r ) (0) = i r x r, so that m nK(t)=-2nt +n E r=so+3 I r =0 for < r < s O + Then tr Y, r +Bexp(-E n 2a') (9.5 1) DERIVATION OF THE BASIC INTEGRAL ... 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 ... 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...
- 13
- 364
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 10 potx
... 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 ... We shall begin along the lines of „„ 3, 9.4 Note the basic equation (9.3 6), and set ° = 2-a Following the arguments of the last chapter we arrive at an analogue of (9.4 4) z n -° exp(nK(t)-n...
- 8
- 304
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 11 doc
... 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 ... 9.11) „ 10 Completion of the proof of Theorem 11 2.1 We now investigate, for < r,
- 28
- 396
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 12 ppsx
... 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...
- 18
- 338
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 13 pdf
... 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...
- 10
- 302
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 14 docx
... 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...
- 13
- 334
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 15 pptx
... 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 ... F ( 15 5) D The left-hand side of (15 5) may be regarded as the greatest distance (in the sense of (15 3)) of the set of n-fold convolutions F§ from the set of infinitely divisible distributions...
- 17
- 331
- 0
Independent And Stationary Sequences Of Random Variables - Chapter 16 ppsx
... family of mappings Tt of9 N 0,, into itself given by the following rule For events A of the form (16.2.1), Tt A= T t {(Xt1 , , Xrs)EA} = {(Xt1+t, , Xrs +t)EA} The set 91 of events of the ... 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 ... case, the set of indicators of events of the form (16.2.1), with A an s-dimensional rectangle with rational vertices, forms a countable set which is contained in no proper closed subspace of Ham...
- 17
- 310
- 0
Tìm thêm: xác định các mục tiêu của chương trình khảo sát chương trình đào tạo của các đơn vị đào tạo tại nhật bản khảo sát chương trình đào tạo gắn với các giáo trình cụ thể xác định thời lượng học về mặt lí thuyết và thực tế tiến hành xây dựng chương trình đào tạo dành cho đối tượng không chuyên ngữ tại việt nam điều tra đối với đối tượng giảng viên và đối tượng quản lí khảo sát thực tế giảng dạy tiếng nhật không chuyên ngữ tại việt nam nội dung cụ thể cho từng kĩ năng ở từng cấp độ xác định mức độ đáp ứng về văn hoá và chuyên môn trong ct phát huy những thành tựu công nghệ mới nhất được áp dụng vào công tác dạy và học ngoại ngữ các đặc tính của động cơ điện không đồng bộ hệ số công suất cosp fi p2 đặc tuyến hiệu suất h fi p2 đặc tuyến tốc độ rôto n fi p2 đặc tuyến dòng điện stato i1 fi p2 động cơ điện không đồng bộ một pha sự cần thiết phải đầu tư xây dựng nhà máy phần 3 giới thiệu nguyên liệu từ bảng 3 1 ta thấy ngoài hai thành phần chủ yếu và chiếm tỷ lệ cao nhất là tinh bột và cacbonhydrat trong hạt gạo tẻ còn chứa đường cellulose hemicellulose chỉ tiêu chất lượng 9 tr 25