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Convolution quotients of nonnegative functions
Abstract
LetG be a locally compact commutative Hausdorff group andf a function belonging toL
1(G). If the integral off with respect to the Haar measure is positive, then one can find a nonnegative (not identically 0) functiong such that the convolution off andg is also nonnegative.
... Next we give our second proof of Theorem 1.2. This proof, which does not involve Fourier analysis, is based on the following result due to Ruzsa and Székely [RS83]. ...
... In fact this is proved in [RS83] not only for functions on R, but on a wider class of locally compact abelian groups, which in particular includes also R d for every d 1. ...
We say that a function tiles at level w by a discrete translation set , if we have a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.
... Finally there is Szekely's "half-coin" interpretation of NP [50,54]. The idea is that two probability distributions that are negative may give rise to a nonnegative proper probability distribution. ...
... The idea is that two probability distributions that are negative may give rise to a nonnegative proper probability distribution. In this interpretation, negative probabilities P are related to a proper probability p via a convolution equation P * p − = p + , which is always possible to be found [50,54]. This convolution means that for a random variable X whose (negative) probability distribution is P , there exists two other random variables, X + and X − with proper probability distributions (p + and p − , respectively) and such that X = X + − X − . ...
There has been a growing interest, both in physics and psychology, in
understanding contextuality in experimentally observed quantities. Different
approaches have been proposed to deal with contextual systems, and a promising
one is contextuality-by-default, put forth by Dzhafarov and Kujala. The goal of
this paper is to present a tutorial on a different approach: negative
probabilities. We do so by presenting the overall theory of negative
probabilities in a way that is consistent with contextuality-by-default and by
examining with this theory some simple examples where contextuality appears,
both in physics and psychology.
... 4. Some bizarre properties of DCP related to signed r.v. Ruzsa and Székely (1983) proves a theorem related to signed random variables and Székely (2005) gives the following result. ...
In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller's characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the L\'evy-Wiener theorem. This is a special case of an open problem which is proposed by Sato(2014), Chaumont and Yor(2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter in the DPCP class cannot be arbitrarily small.
... Is it true that if P is a signed probability distribution such that the probability of the whole space is 1, then we can always find two nonsigned probability distributions, Q and R, such that the convolution of P and Q is R. The answer is: YES (see [16]) and this can be considered a fundamental result for negative probabilities. Directly, we cannot observe a signed probability distribution P that has events with negative probabilities, but if we add a suitable independent error with nonsigned probability distribution Q, then we always get a nonsigned, observable, probability distribution R, which is the convolution of P and Q. ...
... Let us first tackle the issue of meaning. There are some proposals to give meaning to negative probabilities, such as reinterpreting them in terms of violations of the principle of stability for probabilities [32,36,37,38], thinking of them as events that can erase entries in a data table [1,8,7], using an analogy to a half-coin through convolution coefficients [49,54], or perhaps even coming from more ontological principles such as the indistinguishability of fundamental particles [23]. ...
One of the central goals of science is to find consistent and rational representations of observational data: a map of the world, if you will. How we do this depends on the specific tools, which are often mathematical. When dealing with real-world situations, where the data (or “territory”) has a random component, the mathematical tools most commonly used are those grounded in probability theory, defined in a precise way by the Russian mathematician Andrei Kolmogorov. In this paper we explore how experimental data (the “territory”) can be represented (or “mapped”) consistently in terms of probability theory, and present examples of situations, both in the physical and social sciences, where such representations are impossible. This suggests that some “territories” cannot be “mapped” in a way that is consistent with classical logic and probability theory.
... One type of event erases recordings of the other type, and this allows for the observed correlations. Finally, closely related to Abramsky and Brandenburger's, is Szekely's interpretation, who thinks of negative probabilities P as related to a proper probability p via a convolution equation P * f = p, which always exists [80,88]. This convolution means 24 A Dutch Book is the name given to a strategy that would allow one of the gamblers to win for sure over the other gamblers in a game [7]. ...
This review paper has three main goals. First, to discuss a contextual neurophysiologically plausible model of neural oscillators that reproduces some of the features of quantum cognition. Second, to show that such a model predicts contextual situations where quantum cognition is inadequate. Third, to present an extended probability theory that not only can describe situations that are beyond quantum probability, but also provides an advantage in terms of contextual decision-making.
... 4. Some bizarre properties of DCP related to signed r.v. Ruzsa and Székely (1983) proves a theorem related to signed random variables and Székely (2005) gives the following result. ...
In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller's characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the L\'evy-Wiener theorem. This is a special case of an open problem which is proposed by Sato(2014), Chaumont and Yor(2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter in the DPCP class cannot be arbitrarily small.
A generalization of the concept of independence and identical distributiveness of random variables in terms of characteristic functions is presented. We will show that some classical theorems remain valid for these generalizations.
The complete list of prime distributions will be given on locally compact Abelian groups.
Letℱ denote the convolution semigroup of probability distributions on the real line. We prove thatno element of ℱ is prime in the sense that given anℱ one can always find two distributionsG,H∈ℱ such thatF is a convolution factor ofG#x002A;H but neither ofG nor ofH. In contrast,ℱ is known to possess many irreducible elements.
No distribution is prime
- I Z Sz~r~ely
RUZSA, I.Z., SZ~r~ELY, G.J.: No distribution is prime. Preprint.
Asymptotic behavior of products + Cin locally compact spaces
- W R Emerson
- F P Greenleaf
- C + Cp
EMERSON, W.R., GREENLEAF, F.P.: Asymptotic behavior of products CP = C +... + Cin locally compact spaces. Trans. Amer. Math. Soc. 145, 171--204 (1967).
Irreducible and prime distributions In: Probability Measures on Groups
- I.Z. Ruzsa
Classical Harmonic Analysis and Locally Compact Groups
- H Reiter
- H. Reiter
Asymptotic behavior of productsC p =C+
- W R Emerson
- F P Greenleaf
- W.R. Emerson