23,056 research outputs found
Monotone and Boolean Convolutions for Non-compactly Supported Probability Measures
The equivalence of the characteristic function approach and the probabilistic
approach to monotone and boolean convolutions is proven for non-compactly
supported probability measures. A probabilistically motivated definition of the
multiplicative boolean convolution of probability measures on the positive
half-line is proposed. Unlike Bercovici's multiplicative boolean convolution it
is always defined, but it turns out to be neither commutative nor associative.
Finally some relations between free, monotone, and boolean convolutions are
discussed.Comment: 32 pages, new Lemma 2.
Analytic continuations of Fourier and Stieltjes transforms and generalized moments of probability measures
We consider analytic continuations of Fourier transforms and Stieltjes
transforms. This enables us to define what we call complex moments for some
class of probability measures which do not have moments in the usual sense.
There are two ways to generalize moments accordingly to Fourier and Stieltjes
transforms; however these two turn out to coincide. As applications, we give
short proofs of the convergence of probability measures to Cauchy distributions
with respect to tensor, free, Boolean and monotone convolutions.Comment: 13 pages; to appear in Journal of Theoretical Probabilit
Rates of convergence of nonextensive statistical distributions to Levy distributions in full and half spaces
The Levy-type distributions are derived using the principle of maximum
Tsallis nonextensive entropy both in the full and half spaces. The rates of
convergence to the exact Levy stable distributions are determined by taking the
N-fold convolutions of these distributions. The marked difference between the
problems in the full and half spaces is elucidated analytically. It is found
that the rates of convergence depend on the ranges of the Levy indices. An
important result emerging from the present analysis is deduced if interpreted
in terms of random walks, implying the dependence of the asymptotic long-time
behaviors of the walks on the ranges of the Levy indices if N is identified
with the total time of the walks.Comment: 20 page
The Arithmetic of Distributions in Free Probability Theory
We give an analytical approach to the definition of additive and
multiplicative free convolutions which is based on the theory of Nevanlinna and
of Schur functions. We consider the set of probability distributions as a
semigroup equipped with the operation of free convolution and prove a
Khintchine type theorem for the factorization of elements of this semigroup. An
element of contains either indecomposable ("prime") factors or it
belongs to a class, say , of distributions without indecomposable factors.
In contrast to the classical convolution semigroup in the free additive and
multiplicative convolution semigroups the class consists of units (i.e.
Dirac measures) only. Furthermore we show that the set of indecomposable
elements is dense in .Comment: 66 pages; latex; 5 figures; corrected version of proofs of Khintchine
type theorems. For details see end of introductio
Sieving random iterative function systems
It is known that backward iterations of independent copies of a contractive
random Lipschitz function converge almost surely under mild assumptions. By a
sieving (or thinning) procedure based on adding to the functions time and space
components, it is possible to construct a scale invariant stochastic process.
We study its distribution and paths properties. In particular, we show that it
is c\`adl\`ag and has finite total variation. We also provide examples and
analyse various properties of particular sieved iterative function systems
including perpetuities and infinite Bernoulli convolutions, iterations of
maximum, and random continued fractions.Comment: 36 pages, 2 figures; accepted for publication in Bernoull
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