moments of random variables amp 8212 some moment and probability inequalities

... Exercises 105 Moments of Random Variables Some Moment and Probability Inequalities 106 5.1 5.2 5.3 Moments of Random Variables 106 Exercises 111 Expectations and Variances of Some R.V.’s 114 Exercises ... Random 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 ... to skip some fine points of some of the proofs (or some of the proofs altogether!) when studying the book On the other hand, the careful handling of these same fine points should offer some satisfaction...

Ngày tải lên: 09/06/2015, 15:11

... must be false § Moments and characteristic functions Tie moments oc v and absolute moments f3, of a random variable X with di tribution F are defined respectively by 1.4 MOMENTS AND CHARACTERISTIC ... contents and methods of attack of this book, and we seek therefore to so in this preface The problems studied here concern sums of stationary sequences of random variables, including sequences of ... problems of the analytical structure of stable laws The book presupposes a knowledge of the monograph "Limit Distributions of Sums of Independent Random Variables" by B V Gnedenko and A N Kolmogorov,...

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 ... 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 ... 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

... 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 of ξi ... 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 ξη G ξ, η 1.6 Equality holds if and only if P ξ ... 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...

Ngày tải lên: 22/06/2014, 02:20

... must be false § Moments and characteristic functions Tie moments oc v and absolute moments f3, of a random variable X with di tribution F are defined respectively by 1.4 MOMENTS AND CHARACTERISTIC ... expectation of X, and is denoted by the symbol E(X) I f X is a random vector with values in R" and distribution F, and is a Borel measurable function from R" to R, then (X) is a random variable, and ... 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...

Ngày tải lên: 27/06/2014, 03:20

... must be false § Moments and characteristic functions Tie moments oc v and absolute moments f3, of a random variable X with di tribution F are defined respectively by 1.4 MOMENTS AND CHARACTERISTIC ... expectation of X, and is denoted by the symbol E(X) I f X is a random vector with values in R" and distribution F, and is a Borel measurable function from R" to R, then (X) is a random variable, and ... 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...

Ngày tải lên: 02/07/2014, 20:20

... 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 side ... 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 ... Bergstrom [16] and Pollard [135] The special case of the "extreme" stable laws p(x ; a, + 1) is due to Kolmogorov The method of proof is that of contour integration and, later, the technique of steepest...

Ngày tải lên: 02/07/2014, 20:20

... of some stable distribution G with exponent a (0

Ngày tải lên: 02/07/2014, 20:20

... 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, and that ... i = 27r/h and define S (x) and d,,(t) as in § 3.3 Theorem 5.3 If X1 , X2 , are independent random variables with the same distribution F belonging to L h and having finite third moment, then, ... 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 distinguish two cases...

Ngày tải lên: 02/07/2014, 20:20

... system is due to Cramer, STATEMENT OF THE PROBLEM 59 the ai, bj are moments of XX, and collective theorems hold for / ( n) = na (a < Z) If not all the moments of XX exist, limit theorems may still ... centralised and 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 ... 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, for some...

Ngày tải lên: 02/07/2014, 20:20

... 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 ... independent random variables X1 , X2 , satisfy E (Xl) = a1 , V (Xl) _ #j , (7.5.1) and write ZI =XI -a1 , Bn=/31+fl2+ +/3n, Sn = Z I +Z2+ +Zn Theorem 5.1 Suppose that, for some H > and all ... + 3) terms of the expansion of K (zo) - Qrzo = K (zo) - zo K' (zo) in powers of 'L Hence from (7 9) we can determine the cumulants y, n (m < k + 3) and hence the moments /1m (m < k + 3) The...

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

... are equal to the moments of the normal distribution with mean K and variance K Theorem 9.3 follows on recalling that a distribution all of whose moments are the same as those of a given normal ... (9.2.1) holds and that 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)] ... where A < is a constant, and / = 4a/(2a + 1) From (9.3 2) all the moments °k = E X; exist, and there is no loss of generality in taking c =1 Suppose that a < i is fixed and, if a >, 6, take the...

Ngày tải lên: 02/07/2014, 20:20

... (10.1 9) is satisfied (and then by the previous theorem (10 10) 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+ ... Application of the method of steepest descents The integrand of (10 10) is an entire function, since Km (t) is a polynomial Set z=x/n+, 1

Ngày tải lên: 02/07/2014, 20:20

... 16.1) and (11 2) implies the existence of third moments, but not that of moments of all orders In this case it is possible [4] to establish by classical methods that [0, (log n)2 / p (n)] and ... zones of local normal attraction for variables in (d), and of integral normal attraction in general, and that [0, (log n)l p (n) ] and [ - (log n)4 p (n), 0] will not be so unless all the moments ... Let XX be random variables with E (Xi) = 0, V (X;) = 1, and suppose that (11.2.2) is satisfied, where h (x) is a function of Class I We begin by following „ ; determine ° from (11 6) and set ...

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§ ... 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 ~f 3=`r4= where ... the probability density of S;, = ;,1 , X1+ -+X and f, (x) for the probability density of Sn1 = X1+ +Xn l For all we then have f1() = R -1 el (h~)f1() (12.3 7) We shall seek estimates of the...

Ngày tải lên: 02/07/2014, 20:20

... 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 a normal distribution with mean and variance n " ... 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 ... shall need inequalities like (12 2.3) and (12.2.10) for the probability of large deviations We cannot however use the inequalities already proved, since these depended on the vanishing of the first...

Ngày tải lên: 02/07/2014, 20:20

... 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 role of the linear functionals aj , b i is layed by pseudomoments ... the first (h - 1) pseudomoments differ from the corresponding moments only by powers 14 INVESTIGATION OF THE AUXILIARY INTEGRALS 265 of i The pseudomoments play the role of the linear functionals ... 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,...

Ngày tải lên: 02/07/2014, 20:20

... 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

... 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 ~n k=-cc k nx k -C n +1 n there exists one and ... 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 (16.2 ... 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...

Ngày tải lên: 02/07/2014, 20:20