c d f or d f of a random vector amp 8212 basic properties of the d f of a random variable
A Course in Mathematical Statistics
Ngày tải lên :
09/06/2015, 15:11
... Probability of Matchings 47 Exercises 52 The Cumulative Distribution Function (c. d. f or d. f. ) of a Random Vector Basic Properties of the d. f of a Random Variable 85 Exercises 89 The d. f of a Random ... topics are available only in more advanced texts Other features are a complete formulation and treatment of the general Linear Hypothesis and the discussion of the Analysis of Variance as an application ... by developing the fundamentals of modern probability theory and the accompanying mathematical ideas at the beginning of this book so as to prepare the reader for an understanding of the material...
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independent and stationary sequences of random variables
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08/04/2014, 12:28
... a most important fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions ... we 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, ... 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 fundamental conditions Fundamental...
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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
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20/06/2014, 21:20
... be a sequence of independent and identically distributed random variables in the domain of attraction of a normal distribution A universal result in almost sure limit theorem for the self-normalized ... ASCLT for self-normalized partial sums of random variables in the domain of attraction of the normal law, we will show that the ASCLT holds under a fairly general growth condition on dk = k−1 ... 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 1.1 holds...
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Báo cáo hóa học: "Research Article Recurring Mean Inequality of Random Variables" doc
Ngày tải lên :
22/06/2014, 02:20
... Stochastic Orders and Their Applications, Probability and Mathematical Statistics, Academic Press, Boston, Mass, USA, 1994 M Shaked, J G Shanthikumar, and Y L Tong, “Parametric Schur convexity and ... 1, , n Notice that λ1 and λn are the upper and lower bounds of the random variable ζ, so li and Li are the lower and upper bounds of ξi According to Lemma 3.1, we know that A ξ1 , , ξk ... Journal of Inequalities and Applications In addition, if inf ξ ≥ 0, one defines the geometric mean of the random variable ξ, G ξ , to be Gξ sup ξ· inf ξ 1.2 Definition 1.3 If ξ1 , , ξn are bounded...
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Independent And Stationary Sequences Of Random Variables ppt
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27/06/2014, 03:20
... a most important fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions ... we 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, ... positive constants, F a non-decreasing function, G a junction of bounded variation, and f and g their characteristic functions If (1) F( - cc) = G(- co) , F( cc)=G(oo), (2) G'(x) exists for all x and...
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Independent And Stationary Sequences Of Random Variables - Chapter 1 pptx
Ngày tải lên :
02/07/2014, 20:20
... a most important fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions ... positive constants, F a non-decreasing function, G a junction of bounded variation, and f and g their characteristic functions If (1) F( - cc) = G(- co) , F( cc)=G(oo), (2) G'(x) exists for all x and ... convergence of the corresponding functionals, i e F F if and only if (F ,f) - (F, f) for all fe C CONVERGENCE OF DISTRIBUTIONS distribution with mean zero and variance by saying that F -+ F if,...
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Independent And Stationary Sequences Of Random Variables - Chapter 2 ppt
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02/07/2014, 20:20
... we 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, ... distribution for which F" exists, it is unimodal if and only if F" (x) >, for x < a and F" (x) < for x >a The point a is called the mode of the distribution Theorem 2.5.1 If a sequence of unimodal distributions ... 38 STABLE DISTRIBUTIONS Chap Proof Let f be the common characteristic function of the X1 , and let be the characteristic function corresponding to the distribution F Since a degenerate distribution...
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Independent And Stationary Sequences Of Random Variables - Chapter 4 ppt
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02/07/2014, 20:20
... In other words, the theorem shows that, whenever the left-hand side of (3.3.16) has a bounded derivative, that expansion may be formally differentiated We indicate the proof only for the case ... many authors, notably Gnedenko, whose work on the subject was motivated by the work of Khinchin [74] on the analytic foundations of statistical mechanics § Local limit theorems for lattice distributions ... exponent a (0
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Independent And Stationary Sequences Of Random Variables - Chapter 5 pot
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02/07/2014, 20:20
... examples of distributions F in the domain 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 ... which cn = I IF,, - 01 IP converges to zero Theorem can be considered as a generalisation of Theorem in which LP appears in place of Lam Theorems and 3 are analogues of Theorems 3 and 3 Theorem ... ATTRACTION OF STABLE LAWS LP of functions Ilfii,= f J f 141 for which the norm If(x)Ipdx 1p l 00 is finite In § it is shown that the domain of attraction of a stable law is not reduced by replacing...
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Independent And Stationary Sequences Of Random Variables - Chapter 6 pptx
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02/07/2014, 20:20
... limit theorems (via characteristic functions and partial differential equations) are too crude for the derivation of sufficiently general results, and most of the theorems about large deviations are ... tails is closely analogous to the classical problem of characterising the possible limit laws for centralised and normalised sums of independent variables, i e the stable laws And of course there ... linear functions of the distribution F of the variables X; Such a limit theorem will have a collective character, since it will show that all distributions for which these linear functionals have given...
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Independent And Stationary Sequences Of Random Variables - Chapter 7 potx
Ngày tải lên :
02/07/2014, 20:20
... 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 a non-negative Fourier transform h(t), then ... which contradicts the maximality of h Assuming that x > 1, x = o (nZ) and keeping the notation of the previous sections, we find that the saddle point is at z o , determined by (7 2.10) For sufficiently ... characteristic 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...
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Independent And Stationary Sequences Of Random Variables - Chapter 8 pps
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02/07/2014, 20:20
... CRAMER'S INTEGRAL THEOREM ; ITS REFINEMENT BY PETROV Chap 172 „ The introduction of auxiliary random variables Since E (exp a JXX j) < oo , we may write, for Jhj
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Independent And Stationary Sequences Of Random Variables - Chapter 9 ppsx
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02/07/2014, 20:20
... this contradicts the assumption of local where lyII < Since `lJso+3 normal attraction The case of equality in (9.3.3) and the case a < require only straightforward modifications of the arguments ... are zones of local normal attraction for all a
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Independent And Stationary Sequences Of Random Variables - Chapter 10 potx
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02/07/2014, 20:20
... proves Theorem 10.1 „ 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 ... 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 9.2.2 almost ... FORMULATION 10 191 so that relations like (10 1.2) and (10.1 3) imply local normal convergence In the last chapter the zones of local normal convergence were characterised, so that we may take...
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Independent And Stationary Sequences Of Random Variables - Chapter 11 doc
Ngày tải lên :
02/07/2014, 20:20
... establish by classical methods that [0, (log n)2 / p (n)] and [ - (log n)-'/p (n), 0] are zones of local normal attraction for variables in (d) , and of integral normal attraction in general, and ... proceed to the proof that (11 2.2) is sufficient for [0, A( n)/p(n)] and [ - A (n) / p (n), 0] to be zones of local normal attraction Indeed we shall prove more generally the sufficiency of E{expah(jXj ... and integral normal convergence for variables of the class (d) defined in Chapter In terms of the function h (x) we define A (n) by the equation h(n1 A( n)) = {A( n)12 ( 11 1) We shall show that,...
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Independent And Stationary Sequences Of Random Variables - Chapter 12 ppsx
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02/07/2014, 20:20
... continuous-component, in which case 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 ... zones of normal attraction The reason why it is necessary to include A in (12.1 3) is that we have used a change of scale already to set a= We remark that the necessity of (12.1 3) has already ... 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...
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Independent And Stationary Sequences Of Random Variables - Chapter 13 pdf
Ngày tải lên :
02/07/2014, 20:20
... appear in 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 ... that (13 5) holds for the original variables 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 ... is a random variable, independent of the Xj , and having a normal distribution with mean and variance n " Thus E(2n) =0, V(Zn )= 1+6 -2 n " -1 ( 13.2.4) The distribution functions of Z n and...
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Independent And Stationary Sequences Of Random Variables - Chapter 14 docx
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02/07/2014, 20:20
... Here a >, (since the variance exists), the A, are constants, with A a > 0, and s>0 The class of such probability densities we call (A) Such variables have only a finite number of moments, and the ... the distribution of X1 , called pseudomoments This theorem has a collective character since the asymptotic form is determined by a finite number of pseudomoments For x < -1, of course, another analogous ... the contour of integration upwards for t>,0 and downwards for t,
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Independent And Stationary Sequences Of Random Variables - Chapter 15 pptx
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02/07/2014, 20:20
... just the assertion that, for all F, inf IF,,-DI < Cn -3 , ( 15 4) D or since C does not depend on F, sup inf IF,,-DI < Cn - F ( 15 5) D The left-hand side of (15 5) may be regarded as the greatest ... 0, this is non-decreasing and right-continuous If X, Y are independent random variables, the concentration function of X + Y is not greater than that of either of them ... K a class of subsets of 21 that no member of K is contained in any other member Then the number v of members of K does not exceed (n n l) such Proof Among the classes K satisfying the conditions...
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Independent And Stationary Sequences Of Random Variables - Chapter 16 ppsx
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02/07/2014, 20:20
... ) for all fE L2 (dF) and all Borel sets A define c [a, b] In fact it is sufficient to b I (f) = f a f( A) d Y (A) since we can then define IA f( A) dY (A) = I (fx (A) ) We first define I (f) whenf ... integrals of which (16 5.1) is a particular case For a detailed account of such integrals see Chapter IX of [31] A random process Y (A) is said to have orthogonal increments if, for any values 21
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