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.

A HÖLDER INEQUALITY FOR NORMS OF POISSONIAN WICK PRODUCTS

2013 ◽  
Vol 16 (03) ◽  
pp. 1350022 ◽  
Author(s):  
ALBERTO LANCONELLI ◽  
AUREL I. STAN
Keyword(s):  
Hölder Inequality ◽  
Wick Product ◽  
Identity Operator ◽  
Second Quantization ◽  
Charlier Polynomials ◽  
Quantization Operator ◽  
Wick Products

An understanding of the second quantization operator of a constant times the identity operator and the Poissonian Wick product, without using the orthogonal Charlier polynomials, is presented first. We use both understanding, with and without the Charlier polynomials, to prove some inequalities about the norms of Poissonian Wick products. These inequalities are the best ones in the case of L1, L2, and L∞ norms. We close the paper with some probabilistic interpretations of the Poissonian Wick product.

A CHARACTERIZATION OF PROBABILITY MEASURES IN TERMS OF WICK PRODUCT INEQUALITIES

2008 ◽  
Vol 11 (03) ◽  
pp. 377-391 ◽  
Author(s):  
AUREL I. STAN
Keyword(s):  
Random Variable ◽  
Wick Product ◽  
Probability Measures ◽  
Polynomial Functions ◽  
Finite Moments ◽  
Normally Distributed

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds: [Formula: see text] We show that this result can be extended to a random variable X, not necessary Gaussian, having an infinite support and finite moments of all orders, if and only if its Szegö–Jacobi sequence {ωk}k ≥ 1 is super-additive.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

scholarly journals Invariant nuclear space of a second quantization operator

1995 ◽  
Vol 25 (3) ◽  
pp. 541-559
Author(s):  
Hong Chul Chae ◽  
Kenji Handa ◽  
Itaru Mitoma ◽  
Yoshiaki Okazaki
Keyword(s):  
Nuclear Space ◽  
Second Quantization ◽  
Quantization Operator

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.

A note on Gamma Wick products

2018 ◽  
Vol 21 (01) ◽  
pp. 1850004 ◽  
Author(s):  
Florin Catrina ◽  
Aurel I. Stan
Keyword(s):  
Integral Representation ◽  
Random Variables ◽  
Hölder Inequality ◽  
Wick Product ◽  
Wick Products

An integral representation of the Wick product for Gamma distributed random variables, with mean greater than [Formula: see text], is presented first. We use this integral representation to prove a Hölder inequality for norms of Gamma Wick products.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (2) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

Inequalities with application in economic risk analysis

1972 ◽  
Vol 9 (02) ◽  
pp. 441-444 ◽  
Author(s):  
Robert A. Agnew
Keyword(s):  
Risk Analysis ◽  
Lower Bounds ◽  
Generating Function ◽  
Random Variable ◽  
Moment Generating Function ◽  
Upper Bounds ◽  
Economic Risk ◽  
The Moment

Two sharp lower bounds for the expectation of a function of a non-negative random variable are obtained under rather weak hypotheses regarding the function, thus generalizing two sharp upper bounds obtained by Brook for the moment generating function. The application of these bounds to economic risk analysis is discussed.

BAYES' FORMULA FOR SECOND QUANTIZATION OPERATORS

2006 ◽  
Vol 06 (02) ◽  
pp. 245-253 ◽  
Author(s):  
ALBERTO LANCONELLI
Keyword(s):  
Conditional Expectation ◽  
Bayes Rule ◽  
Second Quantization ◽  
Absolutely Continuous ◽  
Bayes Formula ◽  
Conditional Expectations ◽  
Quantization Operator ◽  
Continuous Measures ◽  
The Relationship ◽  
Absolutely Continuous Measures

The Bayes' formula provides the relationship between conditional expectations with respect to absolutely continuous measures. The conditional expectation is in the context of the Wiener space — an example of second quantization operator. In this note we obtain a formula that generalizes the above-mentioned Bayes' rule to general second quantization operators.

Generalization of the New Resonance Theory: Second Quantization Operator, Localization Scheme, and Basis Set

2009 ◽  
Vol 5 (7) ◽  
pp. 1741-1748 ◽  
Author(s):  
Atsushi Ikeda ◽  
Yoshihide Nakao ◽  
Hirofumi Sato ◽  
Shigeyoshi Sakaki
Keyword(s):  
Basis Set ◽  
Second Quantization ◽  
Resonance Theory ◽  
Localization Scheme ◽  
Quantization Operator
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