WICK CALCULUS FOR THE SQUARE OF A GAUSSIAN RANDOM VARIABLE WITH APPLICATION TO YOUNG AND HYPERCONTRACTIVE INEQUALITIES

2012 ◽  
Vol 15 (03) ◽  
pp. 1250018 ◽  
Author(s):  
ALBERTO LANCONELLI ◽  
LUIGI SPORTELLI
Keyword(s):  
Gaussian Measure ◽  
Type Inequality ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Probabilistic Interpretation ◽  
Chi Square ◽  
Wick Calculus

We investigate a probabilistic interpretation of the Wick product associated to the chi-square distribution in the spirit of the results obtained in Ref. 7 for the Gaussian measure. Our main theorem points out a profound difference from the previously studied Gaussian7 and Poissonian12 cases. As an application, we obtain a Young-type inequality for the Wick product associated to the chi-square distribution which contains as a particular case a known Nelson-type hypercontractivity theorem.

THE GROSS DERIVATIVE AND GENERALIZED RANDOM VARIABLES

1999 ◽  
Vol 02 (03) ◽  
pp. 381-396 ◽  
Author(s):  
FRED ESPEN BENTH
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Random Variables ◽  
Random Variable ◽  
Representation Formula ◽  
Gaussian Random Variable ◽  
Wick Product ◽  
Chaos Expansion ◽  
Schwartz Distributions

We extend the Gross derivative to a space of generalized random variables which have a (formal) chaos expansion with kernels from the space of tempered Schwartz distributions. The extended derivative, which we call the Hida derivative, has to be interpreted in the sense of distributions. Many of the properties of the Gross derivative are proved to hold for the extension as well. In addition, we derive a representation formula for the Hida derivative involving the Wick product and a centered Gaussian random variable. We apply our results to calculate the Hida derivative of a class of stochastic differential equations of Wick type.

A generalized Ostrowski type inequality for a random variable whose probability density function belongs to L∞[a, b]

2008 ◽  
Vol 41 (3) ◽  
Author(s):  
Arif Rafiq ◽  
Nazir Ahmad Mir ◽  
Fiza Zafar
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Type Inequality ◽  
Random Variable ◽  
Ostrowski Type Inequality

AbstractWe establish here an inequality of Ostrowski type for a random variable whose probability density function belongs to L

TESTING CHAOS OF TIME SERIES: PORTMANTEAU STATISTICS UNDER AN ORDER TRANSFORMATION

2001 ◽  
Vol 11 (06) ◽  
pp. 1761-1769 ◽  
Author(s):  
DEJIAN LAI
Keyword(s):  
Time Series ◽  
Null Hypothesis ◽  
Random Variable ◽  
Asymptotic Distributions ◽  
Test Statistics ◽  
Chi Square ◽  
Gaussian Series ◽  
Low Dimensional ◽  
Portmanteau Statistics ◽  
Direct Use

This paper studies several portmanteau test statistics with a nonparametric order transformation for distinguishing independent and identically distributed (i.i.d.) random processes from noisy chaotic time series. These portmanteau test statistics are asymptotically distributed as a chi-square random variable under the null hypothesis of i.i.d. Gaussian series. In this Letter, we show that the asymptotic distributions of these portmanteau test statistics on the transformed series are still chi-square under the null hypothesis. The simulations indicate that direct use of these portmanteau test statistics yields low power in identifying chaos. However, with the proposed order transformation, the simulations show that these test statistics are still effective for identifying noisy low dimensional chaos in some cases.

scholarly journals Hölder-Type Inequalities for Norms of Wick Products

2008 ◽  
Vol 2008 ◽  
pp. 1-22 ◽  
Author(s):  
Alberto Lanconelli ◽  
Aurel I. Stan
Keyword(s):  
Random Variable ◽  
Upper Bounds ◽  
Minimal Condition ◽  
Wick Product ◽  
Identity Operator ◽  
Second Quantization ◽  
Measurable Functions ◽  
Quantization Operator ◽  
Wick Products ◽  
Finite Moments

Various upper bounds for the L2-norm of the Wick product of two measurable functions of a random variable X, having finite moments of any order, together with a universal minimal condition, are proven. The inequalities involve the second quantization operator of a constant times the identity operator. Some conditions ensuring that the constants involved in the second quantization operators are optimal, and interesting examples satisfying these conditions are also included.

scholarly journals ASYMPTOTIC CRAMÉR TYPE DECOMPOSITION FOR WIENER AND WIGNER INTEGRALS

2013 ◽  
Vol 16 (01) ◽  
pp. 1350005
Author(s):  
SOLESNE BOURGUIN ◽  
JEAN-CHRISTOPHE BRETON
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Gaussian Random Variable ◽  
Independent Random Variables ◽  
Asymptotic Decomposition ◽  
Type Decomposition

We investigate generalizations of the Cramér theorem. This theorem asserts that a Gaussian random variable can be decomposed into the sum of independent random variables if and only if they are Gaussian. We prove asymptotic counterparts of such decomposition results for multiple Wiener integrals and prove that similar results are true for the (asymptotic) decomposition of the semicircular distribution into free multiple Wigner integrals.

Sign in / Sign up

Export Citation Format

Share Document