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Extremal properties of Rademacher functions with applications to Khintchine and Rosenthal inequalities
 Trans. Amer. Math. Soc
, 1997
"... ��×ØÖ � Ø The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the p moment of a sum of independent symmetric random variables to that of the p and 2 moments of the individual variables, are computed in the range 2
Abstract

Cited by 19 (2 self)
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��×ØÖ � Ø The best constant and the extreme cases in an inequality of H.P. Rosenthal, relating the p moment of a sum of independent symmetric random variables to that of the p and 2 moments of the individual variables, are computed in the range 2 <p≤4. This complements the work of Utev who has done the same for p>4. The qualitative nature of the extreme cases turns out to be different for p<4than for p>4. The method developed yields results in some more general and other related moment inequalities. 1.
Optimal Stopping and Effective Machine Complexity in Learning
 Advances in Neural Information Processing Systems 6
, 1994
"... We study the problem of when to stop learning a class of feedforward networks  networks with linear outputs neuron and fixed input weights  when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there are in g ..."
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We study the problem of when to stop learning a class of feedforward networks  networks with linear outputs neuron and fixed input weights  when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there are in general three distinct phases in the generalization performance in the learning process, and in particular, the network has better generalization performance when learning is stopped at a certain time before the global minimum of the empirical error is reached. A notion of effective size of a machine is defined and used to explain the tradeoff between the complexity of the machine and the training error in the learning process. The study leads naturally to a network size selection criterion, which turns out to be a generalization of Akaike's Information Criterion for the learning process. It is shown that stopping learning before the global minimum of the empirical error has the effect of ne...
1 SCHLÖMILCH AND BELL SERIES FOR BESSEL’S FUNCTIONS, WITH PROBABILISTIC APPLICATIONS.
, 804
"... We have introduced and investigated socalled Schlömilchs and Bell’s series for modified Bessel’s functions, namely, their asymptotic and nonasymptotic properties, connection with Stirling’s and Bell’s numbers etc. We have obtained exact constants in the moment inequalities for sums of centered ind ..."
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We have introduced and investigated socalled Schlömilchs and Bell’s series for modified Bessel’s functions, namely, their asymptotic and nonasymptotic properties, connection with Stirling’s and Bell’s numbers etc. We have obtained exact constants in the moment inequalities for sums of centered independent random variables, improved their asymptotical properties, found lower and upper bounds, calculated a more exact approximation, elaborated the numerical algorithm for their calculation, studied the class of smoothing, etc.
© Hindawi Publishing Corp. MOMENT INEQUALITIES CONNECTED WITH ACCOMPANYING POISSON LAWS IN ABELIAN GROUPS
, 2002
"... We obtain exact inequalities which connect moments of some functions of sums of independent random variables taking values in a measurable Abelian group and those for the accompanying infinitely divisible laws. Some applications to empirical processes are studied. ..."
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We obtain exact inequalities which connect moments of some functions of sums of independent random variables taking values in a measurable Abelian group and those for the accompanying infinitely divisible laws. Some applications to empirical processes are studied.