376,857 research outputs found
Stable laws and domains of attraction in free probability theory
In this paper we determine the distributional behavior of sums of free (in
the sense of Voiculescu) identically distributed, infinitesimal random
variables. The theory is shown to parallel the classical theory of independent
random variables, though the limit laws are usually quite different. Our work
subsumes all previously known instances of weak convergence of sums of free,
identically distributed random variables. In particular, we determine the
domains of attraction of stable distributions in the free theory. These freely
stable distributions are studied in detail in the appendix, where their
unimodality and duality properties are demonstrated.Comment: 38 pages, published versio
Asymptotic tail behavior of phase-type scale mixture distributions
We consider phase-type scale mixture distributions which correspond to
distributions of a product of two independent random variables: a phase-type
random variable and a nonnegative but otherwise arbitrary random variable
called the scaling random variable. We investigate conditions for such a
class of distributions to be either light- or heavy-tailed, we explore
subexponentiality and determine their maximum domains of attraction. Particular
focus is given to phase-type scale mixture distributions where the scaling
random variable has discrete support --- such a class of distributions has
been recently used in risk applications to approximate heavy-tailed
distributions. Our results are complemented with several examples.Comment: 18 pages, 0 figur
Anomalous scaling due to correlations: Limit theorems and self-similar processes
We derive theorems which outline explicit mechanisms by which anomalous
scaling for the probability density function of the sum of many correlated
random variables asymptotically prevails. The results characterize general
anomalous scaling forms, justify their universal character, and specify
universality domains in the spaces of joint probability density functions of
the summand variables. These density functions are assumed to be invariant
under arbitrary permutations of their arguments. Examples from the theory of
critical phenomena are discussed. The novel notion of stability implied by the
limit theorems also allows us to define sequences of random variables whose sum
satisfies anomalous scaling for any finite number of summands. If regarded as
developing in time, the stochastic processes described by these variables are
non-Markovian generalizations of Gaussian processes with uncorrelated
increments, and provide, e.g., explicit realizations of a recently proposed
model of index evolution in finance.Comment: Through text revision. 15 pages, 3 figure
Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions
We study the asymptotic laws for the spatial distribution and the number of
connected components of zero sets of smooth Gaussian random functions of
several real variables. The primary examples are various Gaussian ensembles of
real-valued polynomials (algebraic or trigonometric) of large degree on the
sphere or torus, and translation-invariant smooth Gaussian functions on the
Euclidean space restricted to large domains
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