two functions of two random variables

... inequality functions of associated random variables We now generalise the above results to some nonmonotonic functions of associated random variables In Section 2, we discuss some preliminaries Two ... for nonmonotonic functions of associated random variables in Section As an application, a strong law of large numbers is derived for nonmonotonic functions of associated random variables in Section ... 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 the conditions...

Ngày tải lên: 22/06/2014, 22:20

... of One Random Variable 4.3 Functions of Two Random Variables 4.4 Functions of n Random Variables 4.5 Expectation 4.6 Moment Generating Functions 4.7 Characteristic Functions 4.8 The Laws of Large ... Start of Citation[PU]McGraw-Hill Professional[/PU][DP]1997[/DP]End of Citation Preface The purpose of this book is to provide an introduction to principles of probability, random variables, and random ... Introduction 2.2 Random Variables 2.3 Distribution Functions 2.4 Discrete Random Variables and Probability Mass Functions 2.5 Continuous Random Variables and Probability Density Functions 2.6 Mean...

Ngày tải lên: 31/03/2014, 16:25

... convergence results for independent random variables have been extended to dependent random variables by many authors The concept of negatively associated random variables was introduced by Alam ... version of Theorem 1.2 for negatively dependent random variables Theorem 1.6 Let {Xni, ≤ i ≤ kn, n ≥ 1} be an array of rowwise negatively dependent random variables and {an, n ≥ 1} a sequence of nonnegative ... exists An infinite family of random variables is negatively associated if every finite subfamily is negatively associated The concept of negatively dependent random variables was introduced by...

Ngày tải lên: 20/06/2014, 22:20

... 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 To see how ... right-hand side of 1.4 can be negative, zero, or positive, depending on the dependence structure between the random variables X and Y , and also depending on the monotonicity of functions v and ... have been many attempts to express the covariance Cov X, Y in terms of the cumulative distribution functions of the random variables X and Y Among them is a result by Hoeffding 20 , who proved...

Ngày tải lên: 21/06/2014, 18:20

... case of the Paley-Zygmund inequality: for any nonnegative random variable X, Pr[X > 0] ≥ (EX)2 /EX THE CHROMATIC NUMBER OF A RANDOM GRAPH 1337 Specifically, the number of k-colorings of a random ... therein for accounts of many remarkable results on random graphs, as well as for their connections to various areas of mathematics In the present paper we consider random graphs of bounded average ... has been in the forefront of random graph theory, motivating some of the field’s most significant developments Indeed, one of the most fascinating facts known [13] about random graphs is that for...

Ngày tải lên: 29/03/2014, 07:20

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 ... composed random variables and their stabilities Dotor thesis, Hanoi 2000 [2] Tran Kim Thanh, Nguyen Huu Bao, On the geometric composed variables and the estimate of the stable degree of the Renyi’s ... )|t| = C|t| (1 − |ϕ(t) − 1|) (do |ϕ(t) − 1| µF |t| ∀t) where C is a constant independent of ε and µF = Proof of Theorem 2.1 +∞ −∞ |x|dF (x) < ∞ For every N > and t ∈ R1, we have +∞ |ψ(t) − ψ1 (t)|...

Ngày tải lên: 14/03/2014, 13:20

... 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 ... 171 Statement of the theorem The 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 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 X (t),...

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 ... < 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 ... 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

... 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 completely ... the corresponding results of independent random variable sequences, there still remains much to be desired The concept of complete convergence of a sequence of random variables was introduced ... readability of the article Author details School of Mathematics Science, University of Electronic Science and Technology of China, Chengdu 610054, PR China College of Science, Guilin University of Technology,...

Ngày tải lên: 20/06/2014, 22:20

... is NOD An array of random variables {Xni, i ≥ 1, n ≥ 1} is called rowwise NOD random variables if for every n ≥ 1, {Xni, i ≥ 1} is a sequence of NOD random variables The concept of NOD sequence ... strong law of large numbers for arrays of rowwise NOD random variables Corollary 2.5 Let {Xni, i ≥ 1, n ≥ 1} be an array of rowwise NOD random variables and {an, n ≥ 1} be a sequence of positive ... purpose of this article is to study the strong limit theorems for arrays of rowwise NOD random variables As an application, the Chung-type strong law of large numbers for arrays of rowwise NOD random...

Ngày tải lên: 20/06/2014, 22:20

... set of rd-continuous functions, R {p ∈ R : μ t p t > for denotes the set of all regressive and rd-continuous functions, R 0, ∞ , and N0 {0, 1, 2, } denotes the all t ∈ T}, R denotes the set of ... continuous, and nondecreasing for t, s ∈ Ω The remainder of the proof is similar to that of Theorem 2.3, and we omit it here This completes the proof of Theorem 2.9 Remark 2.10 We note that when T1 19 ... in two independent variables on time scales,” Journal of Mathematical Inequalities, vol 2, no 2, pp 163–184, 2008 11 D R Anderson, “Nonlinear dynamic integral inequalities in two independent variables...

Ngày tải lên: 21/06/2014, 05:20

... Journal of Inequalities and Applications Lemma 2.4 see Let {ξk , k ≥ 1} be a sequence of random variables, uniformly bounded below and with finite variances, and let {dk , k ≥ 1} be a sequence of ... mixing sequence of random variables with zero mean and supn E|Xn |2 δ < ∞ for a certain δ > Assume that 1.5 and 1.6 hold Then lim sup n→∞ |Sn | 2σ0 n ln ln n a.s 2.9 2.2 Proof of Theorem 1.4 From ... 2004 J S Jin, “An almost sure central limit theorem for the product of partial sums of strongly missing random variables, ” Journal of Zhejiang University, vol 34, no 1, pp 24–27, 2007 I Berkes and...

Ngày tải lên: 21/06/2014, 05:20

... this inequality is a consequence of Jensen’s inequality Thus, the proof is completed 4.3 10 Journal of Inequalities and Applications Proof of Propositions Proof of Proposition 2.6 Using Corollary ... 2.33 2.34 Journal of Inequalities and Applications then p1 a1 p2 a2 with equality for a1 pn − an p1 a1 ··· ··· a2 p2 a2 ··· 1 ≥P √ −√ pn an ak , 2.35 an Proof of Lemmas Proof of Lemma 2.2 Since ... proof 5.6 0, g x is increasing, and Proof of Proposition 2.7 Using Corollary 2.5, we need to prove that Qi Rk ak − Qi Qi / Qi Rk Rk R / Qi Rk ak k ≥ 4Qi 3Qi Rk ak − 2Rk ak 2Qi 4Rk 5.7 Journal of...

Ngày tải lên: 21/06/2014, 07:20

... weaker than NA Because of the wide applications of ND random variables, the notions of ND dependence of random variables have received more and more attention recently A series of useful results have ... 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 ... properties of independent or NA random variables to the case of ND variables is highly desirable and of considerably significance in the theory and application Strong convergence is one of the most...

Ngày tải lên: 21/06/2014, 07:20

... ρ-mixing sequences of random variables, ” Chinese Annals of Mathematics Series A, vol 9, no 4, pp 409–412, 1988 Q M Shao, “On the invariance principle for ρ-mixing sequences of random variables, ” Chinese ... respectively, not −q/t in Theorem 2.1 of Kuczmaszewska 17 Now, we give the proof of Theorem 2.1 4 Journal of Inequalities and Applications Proof The conclusion of the theorem is obvious if consider ... array of rowwise ρ-mixing random variables satisfying 2/q supn ∞1 ρn 2i < ∞ for some q ≥ and EXni for all n ≥ 1, i ≥ Let the random variables in i each row be stochastically dominated by a random...

Ngày tải lên: 21/06/2014, 07:20

... Definition 1.4 If ξ1 , , ξn are bounded random variables with inf ξi ≥ 0, i 1, , n, one defines the independent geometric mean of the product of random variables ξ1 , , ξn to be n G ξ1 , ... ξn n A ξi , i G ξ1 , , ξn 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 inf η > 0, then Eξ ·Eη2 A ξ, η ≤ E2 ... ξ· inf ξ 1.2 Definition 1.3 If ξ1 , , ξn are bounded random variables, the independent arithmetic mean of the product of random variables ξ1 , , ξn , A ξ1 , , ξn is given by n i n sup...

Ngày tải lên: 22/06/2014, 02:20

... term Thus p1 of the k s φθ1 , q1 of the k s ϕ1 , , qθ2 of the k s ϕθ2 , r1 of the l s κ1 , , rτ1 of the l s κτ1 , s1 of the l s where ≤ pi ≤ φ1 , , pθ1 of the k s ω1 , , sτ2 of the l ... limit inferior of x is defined as −∞ for each n, then P-lim inf x : −∞; if βn > −∞ for some n, then P-lim inf x : supn {βn } Main result The analysis of double sequences of random variables via ... presented the following two notions of subsequence of a double sequence Definition 2.4 The double sequence y is a double subsequence of the sequence x provided that there exist two increasing double...

Ngày tải lên: 22/06/2014, 03:20

... applications,” Journal of Inequalities in Pure and Applied Mathematics, vol 3, no 4, article 65, pp 1–8, 2002 [11] S S Dragomir and Y.-H Kim, “Some integral inequalities for functions of two variables, ” ... inequalities in two variables and their applications,” Journal of Inequalities and Applications, vol 2005, no 4, pp 347–361, 2005 Kelong Zheng: College of Science, Southwest University of Science ... + ∞ x4 y4 di (x, y,s,t)dt ds ≤ ∞ ui dz wi (z) (2.37) for i = 1, ,n The proof is similar to the argument in the proof of Theorem 2.1 with suitable modification We omit the details here Kelong...

Ngày tải lên: 22/06/2014, 18:20

... fact that addition of independent random variables corresponds to multiplication of characteristic functions If the independent variables XL have respective characteristic functions fi(t), then ... 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 ... 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...

Ngày tải lên: 27/06/2014, 03:20