n functions of n random variables

... strong law of large numbers for nonmonotonic functions of associated random variables Corollary 4.2 Let {Yn , n ≥ 1} be sequence of nonmonotonic functions of associated random variables satisfying ... fact that Var 2Sn = Var Sn − Sn + Sn + Sn = Var Sn − Sn + Var Sn + Sn + 2Cov Sn + Sn , Sn − Sn (3.4) Since Sn + Sn and Sn − Sn are nondecreasing functions of associated random variables, it follows ... {Yn , n ≥ 1} be sequence of nonmonotonic functions of associated random Yn , n ≥ Let {bn , n ≥ 1} be a positive variables as defined in (2.6) Suppose that Yn nondecreasing sequence of real numbers...

Ngày tải lên: 22/06/2014, 22:20

... Chapter Functions of Random Variables, Expectation, Limit Theorems 4.1 Introduction 4.2 Functions of One Random Variable 4.3 Functions of Two Random Variables 4.4 Functions of n Random Variables ... Chapter Random Variables 1 7 8 38 2.1 Introduction 2.2 Random Variables 2.3 Distribution Functions 2.4 Discrete Random Variables and Probability Mass Functions 2.5 Continuous Random Variables and ... n( A2 n A,) = n( A2 n A4) = n( A,) = n( A, n A, n A,) = n( A, n A, n A,) = n( A, n A, n A,) = n( A, n A, n A, n A,) = Thus, Fig 1-13 1.35 Consider the experiment of tossing a fair coin repeatedly and...

Ngày tải lên: 31/03/2014, 16:25

... {Xni, ≤ i ≤ kn, n ≥ 1} be an array of rowwise negatively dependent random variables and {an, n ≥ 1} a sequence of nonnegative constants Suppose that the following conditions hold: (i) ∞ n= 1 an ... {Xni, ≤ i ≤ kn, n ≥ 1} be an array of rowwise negatively associated random variables and {an, n ≥ 1} a sequence of nonnegative constants Suppose that the following conditions hold: ∞ n= 1 an (i) ... following result Theorem 1.3 Let {Xni, ≤ i ≤ kn, n ≥ 1} be an array of rowwise independent mean zero random variables and {an, n ≥ 1} a sequence of nonnegative constants Suppose that the following...

Ngày tải lên: 20/06/2014, 22:20

... Gruss-type ¨ bound for the covariance of two transformed random variables by incorporating a notion of quadrant dependence and also utilizing the idea of constraining the means of the random variables ... dependence structure between the random variables X and Y , and also depending on the monotonicity of functions v and w Our research in this paper is devoted to understanding the covariance Cov ... Coruna, Hong Kong Baptist ˜ University, and the Natural Sciences and Engineering Research Council NSERC of Canada References M Egozcue, L Fuentes Garcia, and W.-K Wong, “On some covariance inequalities...

Ngày tải lên: 21/06/2014, 18:20

... of T lymphocytes and epithelial cells to show that galectin binding to N- glycans on membrane glycoproteins enhances surface residency, and is dependent on N- glycan number (protein encoded) and ... dependent on branched N- glycans, and an important aspect of this dependence is galectin binding It is worth noting that fibronectin polymerization and tumor cell motility are regulated by binding ... explain why bisecting GlcNAc-containing N- glycans are abundant in the brain [65] In fact, mice carrying an inactive GnT-III mutant have an atypical neurological phenotype [66] The data obtained in...

Ngày tải lên: 18/02/2014, 17:20

Ngày tải lên: 17/12/2013, 15:18

... occasion of 50th Anniversary of Department of Mathematics, Mechanics and Informatics, Vietnam National University References [1] Tran Kim Thanh, On the characterization of the distribution of the ... constant independent of ε Indeed, by using Essen’s inequality (see [3]) we have 1 ρ(Ψ; Ψ1 ) C5 ε + C6 ε K1 ε T where K1 is a constant independent of ε (26) C4 ε (27) Nguyen Huu Bao / VNU Journal ... Then we have ρ(Ψ, Ψ1 ) K1 ε1/6 where K1 is a constant independent of ε, Ψ(x) and Ψ (x) are the distribution function of η and η respectively π ; ρ Then we have the Lemma 2.1 Let a is a complex number,...

Ngày tải lên: 14/03/2014, 13:20

... AND Yu V LINNIK University of ' Leningrad, Leningrad INDEPENDENT AND STATIONARY SEQUENCES OF RANDOM VARIABLES Edited by PROFESSOR J F C KINGMAN University of Oxford, Oxford, U K 12240 N" N ... conditional probabilities and expectations 17 Distributions and distribution functions 19 Convergence of distributions 21 Moments and characteristic functions 24 Continuity of the correspondence ... extensions Investigation of the fundamental integral 260 Investigation of the auxiliary integrals 263 265 An example Chapter 15 Approximation of distributions of sums of independent components...

Ngày tải lên: 08/04/2014, 12:28

... 1−α α , ln ln Dn ∼ α ln ln n (11) Thus combining |ξk | ≤ c for any k, n T n3 ≤ c dk k=1 dj 1≤k< j n; j/k 0, note that n ... n ln n exp(lnα x) dx = exp(yα )dy x ln n − α −α ∼ exp(yα ) + y exp(yα ) dy α Dn ∼ ln n = 1−α y exp(yα ) dy α = ln1−α n exp(lnα n) , n → ∞ α This implies ln Dn ∼ lnα n, exp(lnα n) ∼ αDn (ln Dn )...

Ngày tải lên: 20/06/2014, 21:20

... nondecreasing (or non-increasing) functions, then f(X) and g(Y ) are NQD Lemma 3.2 [5] Let {Xn; n ≥ 1} be a sequence of pairwise NQD random variables with EXn = and EXn < ∞ for all n ≥ Then n ... will mean an ≤ c(bn), and an ≪ bn will mean an = O(bn) We will use the following concept in this article Let {Xn; n ≥ 1} be a sequence of NQD random variables and let X be a nonnegative random variable ... establishing almost sure convergence of summation of random variables Hsu and Robbins [8] proved that the sequence of arithmetic means of independent and identically distributed random variables converges...

Ngày tải lên: 20/06/2014, 22:20

... n ≤C n= 1 i=1 ∞ n= 1 i=1 n |Egn (|Xni |) +C gn (an ) n= 1 ∞ Egn (|Xni |) +C gn (an ) n= 1 Egn (|Xni |) < ∞, gn (an ) i=1 n i=1 E|Xni |2 I(|Xni | ≤ an ) a2 n E|Xni |2 I(|Xni | ≤ an ) a2 n Egn (|Xni ... I(|Xni | ≤ an ) an E i=1 β |Xni | E β an i=1 n n I(|Xni | > an ) + Xni an +C Egn Xni an +C i=1 n ≤C i=1 β an i=1 n Egn ≤C |Xni |β E Egn Egn 1/β Xni an Xni an i=1 n i=1 1/β I(|Xni | ≤ an ) I(|Xni ... >0 and (2.10), we can get that ∞ P n= 1 (n) Tn ∞ > ε ≤C n= 1 ∞ n n P (|Xni | > an ) + C i=1 ∞ n= 1 i=1 n ≤C+C E n= 1 i=1 ∞ n |Xni |β β an Egn Xni an Egn ≤C+C Xni an n= 1 i=1 ∞ n ≤C+C n= 1 i=1 E|Xni...

Ngày tải lên: 20/06/2014, 22:20

... Applications 2/β max{l : k ≤ l ≤ n, l/k < ln Dn n T2 ,n, 2,2 ≤ n0 n0 n α dk dl ≤ e ln n }, then l 2k dk k l 2k k α l e ln n α n dk ln n0 − ln 2k 2.32 k Dn ln1−1/α Dn ln ln Dn e ln n Dn ln ln Dn By ... mean and unit variance Let ak ,n bk ,n / n ; note that n bk ,n n k−1 b1 ,n k i k n k b1 ,n k k−1 k 2n − b1 ,n , n ≥ 1, 2.10 and via 1.7 we have n sup n k a2 k ,n max |ak ,n | 1≤k n n sup bk ,n n k n bk ,n ... Var n k bk ,n Yk n 1, 2.12 Journal of Inequalities and Applications and applying 1.5 ∞ ∞ n2 /δ α n n n r < ∞ 2.13 as n −→ ∞, 2.14 2/δ n Consequently using Lemma 2.1, we can obtain Sn ,n d − − N 0,...

Ngày tải lên: 21/06/2014, 05:20

... NA implies ND from the definition of NA and ND But ND does not imply NA, so ND is much weaker than NA Because of the wide applications of ND random variables, the notions of ND dependence of random ... 2 Journal of Inequalities and Applications variables Consequently, the following definition is needed to define sequences of negatively dependent random variables Definition 1.2 The random variables ... covariance exists An infinite sequence of random variables {Xn ; n ≥ 1} is said to be NA if every finite subfamily is NA The definition of PND is given by Lehmann , the concept of ND is given by Bozorgnia...

Ngày tải lên: 21/06/2014, 07:20

... Then for any ε > and αp ≥ ⎧ ⎨ n p−2 P max ⎩ 1≤i n ∞ n nαp−2 , bn Proof Put cn ∞ n cn bn i ∞ n p−2 n n −q/t c n bn ≤C n n ⎭ < ∞ n αp |ani |p E|Xni |p ≤ C ∞ n 1 max|ani |p E|Xni |p ≤ C n 1≤i n 3.2 ... q/2 l n max1≤i n |ani |2 E|Xni |2 I |ani Xni | < n1 /t < ∞, then ∞ n l n P n Proof Let cn ⎧ ⎨ max ⎩ 1≤i n i anj Xnj − anj EXnj I anj Xnj < n1 /t > n1 /t j n l n and bn ⎫ ⎬ ⎭ < ∞ 3.13 n Using Theorem ... < and real number λ, and any ε > the following conditions are fulfilled: A ∞ n nλ l n n i B ∞ n nλ−q/t l n max1≤i n E|ani Xni |q I |ani Xni | < n1 /t < ∞, C ∞ n nλ−q/t q/2 P {|ani Xni | ≥ n1 /t...

Ngày tải lên: 21/06/2014, 07:20

... , n are bounded random variables with inf ξi ≥ 0, i 1, , n, one defines the independent geometric mean of the product of random variables ξ1 , , n to be n G ξ1 , , n sup ξi inf ξi ... n are independent, then A ξ1 , , n n A ξi , i G ξ1 , , n 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 ... bounded random variables, the independent arithmetic mean of the product of random variables ξ1 , , n , A ξ1 , , n is given by n i n sup ξi i inf ξ A ξ1 , , n 1.3 Definition 1.4 If...

Ngày tải lên: 22/06/2014, 02:20

... and B such that k,l>B |am ,n, k,l | < A Following Robison and Hamilton work, Patterson in presented the following two notions of subsequence of a double sequence Definition 2.4 The double sequence ... some n, then P-lim sup x : infn { n } Similarly, let n follows: if n infn {xk,l : k, l ≥ n} Then the Pringsheim limit inferior of x is defined as −∞ for each n, then P-lim inf x : −∞; if n > ... > −∞ for some n, then P-lim inf x : supn { n } Main result The analysis of double sequences of random variables via four-dimensional matrix transformations begins with the following theorem However,...

Ngày tải lên: 22/06/2014, 03:20

... important fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions fi(t), ... 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 sense Theorem ... distribution For every n, the sum Xl + 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...

Ngày tải lên: 27/06/2014, 03:20

... important fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions fi(t), ... 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 sense Theorem ... functional (F, f) = f f (x) dF (x) in the space C of continuous functions with limits at oo Weak convergence of distributions is equivalent to weak convergence of the corresponding functionals,...

Ngày tải lên: 02/07/2014, 20:20

... and, for large y > 0, take n so that Bnx

Ngày tải lên: 02/07/2014, 20:20

... written n = mN + r, where m and r are integers and < r < N By Lemma 4.4.2, F,,(x) = F*" (B n x + A n B n) = Bnx+AnBn-ANBN BN + bH2 {aHl Bnx+AnBn-ANBNI BN *m * * F *r (Bn X +An Bn) = ajbm-jH*j B n ... the conditions of Theorem be fulfilled Then, with the same choice of normalising constants A n , Bn , Jim Z JP(k) n - g an B C n + -An B n = (4 12) Proof Denote by G n (x) the distribution on the ... function of the normalised sum Z,, = (X + X2 + + Xn - An)/ B,, Then Fn has a similar decomposition Fn (x) = an R n (x) + b n S n (x) (4.4.2) into absolutely continuous and singular components, and...

Ngày tải lên: 02/07/2014, 20:20