on random variables and their distributions

... Variables and Their Distributions 53 3.1 3.2 3.3 3.4 3.5* Chapter Some General Concepts 53 Discrete Random Variables (and Random Vectors) 55 Exercises 61 Continuous Random Variables (and Random Vectors) ... 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 Vector and Its Properties—Marginal and Conditional d.f.’s and ... sections characteristic functions for the one-dimensional and the multidimensional case, and also by isolating in a section by itself definitions and results on momentgenerating functions and factorial...

Ngày tải lên: 09/06/2015, 15:11

... Distributions and distribution functions 19 Convergence of distributions 21 Moments and characteristic functions 24 Continuity of the correspondence between distributions and characteristic functions ... (b) - F (a), and X (w) = w 21 CONVERGENCE OF DISTRIBUTIONS Let X and Y be independent random variables with respective distribution functions F1 and F2 The distribution function F of X + ... shows that only infinitely divisible distributions can arise as limits of distributions of sums of independent random variables Consider, for each n, a collection of independent random variables, ...

Ngày tải lên: 08/04/2014, 12:28

... that the random variable X belongs to the domain of attraction of the normal law, if there exist constants an > 0, bn ∈ R such that S n − bn d −→ N, an where N is the standard normal random variable ... obtained by Lacey and Philipp [4], Ibragimov and Lifshits [5], Miao [6], Berkes and Cs´ ki [7], H¨ rmann [8], Wu [9, 10], and Ye and Wu [11] Huang and Zhang [12] and Zhang a o and Yang [13] obtained ... ) (14) (15) (16) where dk and Dn are defined by (4) and f is a non-negative, bounded Lipschitz function Proof ¯ By the cental limit theorem for i.i.d random variables and VarS n ∼ nl(ηn ) as n...

Ngày tải lên: 20/06/2014, 21:20

... 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 random variables, ... 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 ... We define the random variable ζ, and assign P ζ λi ζβi /2 , i 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...

Ngày tải lên: 22/06/2014, 02:20

... (b) - F (a), and X (w) = w 21 CONVERGENCE OF DISTRIBUTIONS Let X and Y be independent random variables with respective distribution functions F1 and F2 The distribution function F of X + ... shows that only infinitely divisible distributions can arise as limits of distributions of sums of independent random variables Consider, for each n, a collection of independent random variables, ... and characteristic functions The correspondence between probability distributions on the real line and their characteristic functions is not only one-to-one, but also continuous in the following...

Ngày tải lên: 27/06/2014, 03:20

... (b) - F (a), and X (w) = w 21 CONVERGENCE OF DISTRIBUTIONS Let X and Y be independent random variables with respective distribution functions F1 and F2 The distribution function F of X + ... shows that only infinitely divisible distributions can arise as limits of distributions of sums of independent random variables Consider, for each n, a collection of independent random variables, ... and characteristic functions The correspondence between probability distributions on the real line and their characteristic functions is not only one-to-one, but also continuous in the following...

Ngày tải lên: 02/07/2014, 20:20

... 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 ... functions Finally in § conditions on the distribution of the Xi are given which ensure convergence of the distribution of the normed sums (2.1 3) to a given stable distribution § Canonical ... (2.2.10) Since M is non-decreasing, m is non-increasing, and so therefore is the function defined on the positive rationals Consequently, has right and left limits (s - 0) and (s + 0) at all s...

Ngày tải lên: 02/07/2014, 20:20

... stable distribution G with exponent a (0

Ngày tải lên: 02/07/2014, 20:20

... distribution function of the degenerate distribution concentrated at x=0 From (5.2 2) the function 01 (t) = {fn(t)-1}/it belongs to LP so long as < p < (1- a) -1 , and I I/nJ Ip is bounded by a constant ... 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 distribution G with exponent ... independent random variables with the same distribution F If it is possible to select normalising constants A,,, B„ in such a way that the distributions F,, Qf Zn=(Xi+X2+ + X,,-A„)/B,, (5.2.1) converge...

Ngày tải lên: 02/07/2014, 20:20

... very useful, and is often much easier to compute than the exact expression (6 5) Suppose that the random variables X; introduced at the beginning of the section satisfy Cramer's condition that, ... functions and partial differential equations) are too crude for the derivation of sufficiently general results, and most of the theorems about large deviations are proved under very stringent conditions ... 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 parameters,...

Ngày tải lên: 02/07/2014, 20:20

... sE Because of condition (C) in (7 2) M (z) has an analytic continuation to the strip IRe z( < a, which has a power series expansion about z = convergent in tzI < 2a-a The integrand in (7.2.4) ... 4) Here p0(x)=(27r) 1e-'x' and 2(z)=2 o +2 l z+2 2 + is Cramer's power series, convergent for Izl < E (a), where e (a) depends only on a (cf (6.1 11)) The construction of this power series will ... bounded continuous function g (x) E L (- oo, oo) has a non-negative Fourier transform h(t), then h(t)EL (-co, oc) The relation (7 2.1) permits us to express p" (x) (n>,2) by the inversion formula...

Ngày tải lên: 02/07/2014, 20:20

... 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

... and p (n), p (n), p (n) are positive functions converging to 0o as n-* oo In this chapter we study monomial zones of local normal attraction, both narrow and wide „ The fundamental conditions ... /3 < and (9 2.5) contradicts (9.2.4) 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 [- ... no'p (n) ] and [ - n" p (n), 0] to be zones of local normal attraction, is also sufficient for [0, n"/p (n)] and [ - n"/p (n), 0] to be zones of local normal attraction If on the other hand < a...

Ngày tải lên: 02/07/2014, 20:20

... 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, ... linear functionals aj, bi We see that, in the monomial zones, the only possible limiting tails are those determined by the segments of Cramer's series ~(z) Since s-+ 00 as a > 2, the only possible ... 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 OF THE FUNDAMENTAL...

Ngày tải lên: 02/07/2014, 20:20

... ofTheorems 11 2.1 and 11 2.2 continue to hold „ On the conditions imposed upon h(x) We first comment on the conditions which the different classes, in particular Class I, impose on h (x) The inequality ... (n)] and [ - nZ - R/p (n) ] are zones of local normal attraction, and completes the proof of Theorem 11 2.1 „ 11 The corresponding integral theorem Consider first the monomial zones [0, n"] and ... is natural to assume h to be monotonic and differentiable, so that H is also If we also assume that h' is monotonic, this leads to (11 2) and, in view of the left-hand inequality of (11 1), to...

Ngày tải lên: 02/07/2014, 20:20

... 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§ + Yn) Write ... (t)" On (t) exp (- n ` itx) dt Using the notation of Chapter 9, and following the calculations which there led to (9 5.2), we find that 242 INTEGRAL NORMAL ATTRACTION : WIDE MONOMIAL ZONES ... with those of a normal distribution These conditions are moreover sufficient for [0, n"/ p (n)] and [ - n"/ p (n), 0] to be zones of normal attraction The reason why it is necessary to include...

Ngày tải lên: 02/07/2014, 20:20

... this assertion, we consider the normalised sum (13 2.2) Zn = Sn/6n- , and a modified random variable Z n = Zn + Yn /6n ( 13 3) , where Yn is a random variable, independent of the Xj , and having ... 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 n will be denoted by Fn (x) and Fn (x) respectively, and their characteristic ... independent and identically distributed, with E(Xj)=0, V(Xj)=6 , and suppose that (13 1) is satisfied We prove that, for any monotonic function p ( n) -3 oo there exist positive constants c and c2...

Ngày tải lên: 02/07/2014, 20:20

... integral (14.3 8) the integrand is rational, non-zero on the real axis, has poles „ i and is of order O (~-"`) at infinity Moving the contour of integration upwards for t>,0 and downwards for t,, if is it defined in some neighbourhood [ - to , t o] of t = and coincides with (t) on [0,...

Ngày tải lên: 02/07/2014, 20:20

... Concentration functions The concentration function of a random variable X is the function QX (l) = Q(l) =sup P(x

Ngày tải lên: 02/07/2014, 20:20

... the random variables _ j ~n k=-cc k nx k -C n +1 n there exists one and only one transformation Ti (up to events of probability 0), satisfying the conditions (1 )-{41 ) If now c is any random ... continuous process may be modified to satisfy this condition without altering the finite-dimensional distributions [31] Example The sequence of independent, identically distributed random variables ... 3) Consequently, every continuous positive-definite function is the autocovariance function of some stationary process, and every bounded non-decreasing function is the spectral function of...

Ngày tải lên: 02/07/2014, 20:20