Moment Inequalities via Martingale Methods

2019 ◽  
pp. 62-94
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Type Inequality ◽  
Random Variables ◽  
Maximal Inequality ◽  
Partial Sums ◽  
Moment Inequalities ◽  
Exponential Inequalities ◽  
Conditional Expectations ◽  
Rosenthal Type Inequality ◽  
The Moment ◽  
The Individual

In this chapter we establish different kinds of moment inequalities for partial sums and the maximum of partial sums of a large class of random variables, including martingale sequences, mixingales, and other dependent structures. All the bounds involve the moments of the conditional expectations of either the partial sums or the individual random variables. In most of the proofs martingale approximations are used. This method allows us to use the moment inequalities for the martingale part developed in Chapter 2. We start with a dyadic scheme useful for analysis of the variance of partial sums in the stationary setting. Then, we obtain Burkholder-type inequalities via Maxwell–Woodroofe-type characteristics and an extension of Doob’s maximal inequality for adapted sequences. A Rosenthal-type inequality for stationary sequences is also provided with bounds using conditional expectations of the partial sums. Maximal exponential inequalities are established involving either Maxwell–Woodroofe-type characteristics or the projective operators.

Moment Inequalities and Gaussian Approximation for Martingales

2019 ◽  
pp. 27-61
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Functional Form ◽  
Gaussian Approximation ◽  
Sufficient Conditions ◽  
Maximal Inequality ◽  
Partial Sums ◽  
Moment Inequalities ◽  
Exponential Inequalities ◽  
Moderate Deviations Principle ◽  
Triangular Arrays ◽  
Burkholder Inequality

The aim of this chapter is to present useful tools for analyzing the asymptotic behavior of partial sums associated with dependent sequences, by approximating them with martingales. We start by collecting maximal and moment inequalities for martingales such as the Doob maximal inequality, the Burkholder inequality, and the Rosenthal inequality. Exponential inequalities for martingales are also provided. We then present several sufficient conditions for the central limit behavior and its functional form for triangular arrays of martingales. The last part of the chapter is devoted to the moderate deviations principle and its functional form for triangular arrays of martingale difference sequences.

scholarly journals Maxima of Sums of Heavy-Tailed Random Variables

2002 ◽  
Vol 32 (1) ◽  
pp. 43-55 ◽  
Author(s):  
K.W. Ng ◽  
Q.H. Tang ◽  
H. Yang
Keyword(s):  
Asymptotic Properties ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Tail Probabilities ◽  
Large Classes ◽  
Heavy Tailed Distributions ◽  
Heavy Tailed ◽  
The Individual ◽  
Risk Processes

AbstractIn this paper, we investigate asymptotic properties of the tail probabilities of the maxima of partial sums of independent random variables. For some large classes of heavy-tailed distributions, we show that the tail probabilities of the maxima of the partial sums asymptotically equal to the sum of the tail probabilities of the individual random variables. Then we partially extend the result to the case of random sums. Applications to some commonly used risk processes are proposed. All heavy-tailed distributions involved in this paper are supposed on the whole real line.

Some Inequalities of Partial Sums for Negatively Dependent Random Variables

2012 ◽  
Vol 195-196 ◽  
pp. 694-700
Author(s):  
Hai Wu Huang ◽  
Qun Ying Wu ◽  
Guang Ming Deng
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Exponential Inequalities ◽  
Dependent Random Variables ◽  
Probability Inequalities ◽  
Associated Random Variables ◽  
Negatively Dependent Random Variables

The main purpose of this paper is to investigate some properties of partial sums for negatively dependent random variables. By using some special numerical functions, and we get some probability inequalities and exponential inequalities of partial sums, which generalize the corresponding results for independent random variables and associated random variables. At last, exponential inequalities and Bernsteins inequality for negatively dependent random variables are presented.

scholarly journals Complete convergence for weighted sums of a class of random variables

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 509-522 ◽  
Author(s):  
Xin Deng ◽  
Meimei Ge ◽  
Xuejun Wang ◽  
Yanfang Liu ◽  
Yu Zhou
Keyword(s):  
Complete Convergence ◽  
Type Inequality ◽  
Random Variables ◽  
Random Variable ◽  
Weighted Sums ◽  
Real Numbers ◽  
Mild Conditions ◽  
Rosenthal Type Inequality

Let {ani,1?i?n,n?1} be an array of real numbers and {Xn,n?1} be a sequence of random variables satisfying the Rosenthal type inequality, which is stochastically dominated by a random variable X. Under mild conditions, we present some results on complete convergence for weighted sums ?ni=1 aniXi of random variables satisfying the Rosenthal type inequality. The results obtained in the paper generalize some known ones in the literatures.

scholarly journals Moment inequality of the minimum for nonnegative negatively orthant dependent random variables

Filomat ◽  
2014 ◽  
Vol 28 (7) ◽  
pp. 1475-1481
Author(s):  
Xuejun Wang ◽  
Shijie Wang ◽  
Shuhe Hu
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Exponential Inequality ◽  
Moment Inequalities ◽  
Moment Inequality ◽  
Dependent Random Variables ◽  
The Moment

Let {xn,n ? 1} be a sequence of positive numbers and {?n,n ? 1} be a sequence of nonnegative negatively orthant dependent (NOD) random variables satisfying certain distribution conditions. An exponential inequality for the minimum min1?i?n xi?i is given. In addition, the moment inequalities of the minimum (Ek - min1?i?n|xi?i|p)1/p for nonnegative negatively orthant dependent random variables are established, where p > 0 and k = 1,2,..., n. Our results generalize the corresponding ones for independent random variables to the case of negatively orthant dependent random variables.

Complete convergence and complete moment convergence for extended negatively dependent random variables

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1381-1394 ◽  
Author(s):  
Aiting Shen ◽  
Yu. Zhang ◽  
Wenjuan Wang
Keyword(s):  
Complete Convergence ◽  
Type Inequality ◽  
Random Variables ◽  
Weighted Sums ◽  
Complete Moment Convergence ◽  
Moment Inequalities ◽  
Dependent Random Variables ◽  
Moment Convergence ◽  
Negatively Dependent Random Variables ◽  
End Random Variables

In this paper, we provide some probability and moment inequalities (especially the Marcinkiewicz-Zygmund type inequality) for extended negatively dependent (END, in short) random variables. By using the Marcinkiewicz-Zygmund type inequality and the truncation method, we investigate the complete convergence for sums and weighted sums of arrays of rowwise END random variables. In addition, the complete moment convergence for END random variables is obtained. Our results generalize and improve the corresponding ones of Wang et al. [18] and Baek and Park [2].

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