scholarly journals Moment Inequalities for Maxima of Partial Sums in Probability with Applications in the Theory of Orthogonal Series

2014 ◽  
Vol 61 (06) ◽  
pp. 576 ◽  
Author(s):  
Ferenc Móricz
Keyword(s):  
Orthogonal Series ◽  
Partial Sums ◽  
Moment Inequalities

On the order of partial sums of general orthogonal series

1969 ◽  
Vol 6 (4) ◽  
pp. 725-732
Author(s):  
R. S. Davtyan
Keyword(s):  
Orthogonal Series ◽  
Partial Sums

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.

On a Problem of Littlewood

1996 ◽  
Vol 3 (2) ◽  
pp. 121-132
Author(s):  
M. Khazaradze
Keyword(s):  
Fourier Series ◽  
Trigonometric Series ◽  
Orthogonal Series ◽  
Orthonormal System ◽  
Partial Sums

Abstract The theorem on the tending to zero of coefficients of a trigonometric series is proved when the L 1-norms of partial sums of this series are bounded. It is shown that the analog of Helson's theorem does not hold for orthogonal series with respect to the bounded orthonormal system. Two facts are given that are similar toWeis' theorem on the existence of a trigonometric series which is not a Fourier series and whose L 1-norms of partial sums are bounded.

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