scholarly journals A new characterization of characteristic functions of absolutely continuous distributions

1975 ◽  
Vol 58 (2) ◽  
pp. 323-329 ◽  
Author(s):  
Simeon Berman
Keyword(s):  
Characteristic Functions ◽  
Continuous Distributions ◽  
Absolutely Continuous

scholarly journals A four-dimensional random motion at finite speed

2006 ◽  
Vol 43 (4) ◽  
pp. 1107-1118 ◽  
Author(s):  
Alexander D. Kolesnik
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Transition Density ◽  
Random Motion ◽  
Characteristic Functions ◽  
Conditional Distributions ◽  
Continuous Component ◽  
Absolutely Continuous ◽  
Finite Speed ◽  
Uniform Law

We consider the random motion of a particle that moves with constant finite speed in the space ℝ4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x ∈ ℝ4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.

Entropy of a Convolution Operator

2004 ◽  
Vol 11 (01) ◽  
pp. 79-85 ◽  
Author(s):  
Aleksander Urbański
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Convolution Operator ◽  
Absolutely Continuous ◽  
Doubly Stochastic ◽  
Doubly Stochastic Operator ◽  
Zero Entropy

The concept of the entropy of a doubly stochastic operator was introduced in 1999 by Ghys, Langevin, and Walczak. The idea was developed further by Kamiński and de Sam Lazaro, who also conjectured that the entropy of a convolution operator determined by a probability measure on a compact abelian group is equal to zero. We prove that this is true when the group is connected and the convolution operator is determined by a measure absolutely continuous with respect to the normalized Haar measure. Our result provides also a characterization of the set of doubly stochastic operators with non-zero entropy.

scholarly journals PRESBURGER ARITHMETIC, RATIONAL GENERATING FUNCTIONS, AND QUASI-POLYNOMIALS

2015 ◽  
Vol 80 (2) ◽  
pp. 433-449 ◽  
Author(s):  
KEVIN WOODS
Keyword(s):  
Generating Functions ◽  
Counting Function ◽  
Order Theory ◽  
Characteristic Functions ◽  
Presburger Arithmetic ◽  
First Order ◽  
First Order Theory ◽  
Rational Generating Function ◽  
Rational Generating Functions

AbstractPresburger arithmetic is the first-order theory of the natural numbers with addition (but no multiplication). We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= (p1, . . . ,pn) are a subset of the free variables in a Presburger formula, we can define a counting functiong(p) to be the number of solutions to the formula, for a givenp. We show that every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (02) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

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