392 research outputs found
Some two-dimensional extensions of Bougerol's identity in law for the exponential functional of linear Brownian motion
We present a two-dimensional extension of an identity in distribution due to
Bougerol \cite{Bou} that involves the exponential functional of a linear
Brownian motion. Even though this identity does not extend at the level of
processes, we point at further striking relations in this direction
Level crossings and other level functionals of stationary Gaussian processes
This paper presents a synthesis on the mathematical work done on level
crossings of stationary Gaussian processes, with some extensions. The main
results [(factorial) moments, representation into the Wiener Chaos, asymptotic
results, rate of convergence, local time and number of crossings] are
described, as well as the different approaches [normal comparison method, Rice
method, Stein-Chen method, a general -dependent method] used to obtain them;
these methods are also very useful in the general context of Gaussian fields.
Finally some extensions [time occupation functionals, number of maxima in an
interval, process indexed by a bidimensional set] are proposed, illustrating
the generality of the methods. A large inventory of papers and books on the
subject ends the survey.Comment: Published at http://dx.doi.org/10.1214/154957806000000087 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws
This paper collects facts about the number of occupied boxes in the classical
balls-in-boxes occupancy scheme with infinitely many positive frequencies:
equivalently, about the number of species represented in samples from
populations with infinitely many species. We present moments of this random
variable, discuss asymptotic relations among them and with related random
variables, and draw connections with regular variation, which appears in
various manifestations.Comment: Published at http://dx.doi.org/10.1214/07-PS092 in the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Limit theorems for stationary Markov processes with L2-spectral gap
Let be a discrete or continuous-time Markov process
with state space where is an arbitrary measurable set. Its
transition semigroup is assumed to be additive with respect to the second
component, i.e. is assumed to be a Markov additive
process. In particular, this implies that the first component
is also a Markov process. Markov random walks or additive functionals of a
Markov process are special instances of Markov additive processes. In this
paper, the process is shown to satisfy the following classical
limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c)
the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order
Edgeworth expansion, provided that we have sup{t\in(0,1]\cap T : E{\pi,0}[|Y_t|
^{\alpha}] < 1 with the expected order with respect to the independent case (up
to some for (c) and (d)). For the statements (b) and (d), a
Markov nonlattice condition is also assumed as in the independent case. All the
results are derived under the assumption that the Markov process has an invariant probability distribution , is stationary and has the
-spectral gap property (that is, (X_t)t\in N} is -mixing in
the discrete-time case). The case where is non-stationary is
briefly discussed. As an application, we derive a Berry-Esseen bound for the
M-estimators associated with -mixing Markov chains.Comment: 35 pages Accepted(6 january 2011) for publication in Annales de
l'Institut Henri Poincare - Probabilites et Statistique
A Probabilistic Approach to Generalized Zeckendorf Decompositions
Generalized Zeckendorf decompositions are expansions of integers as sums of
elements of solutions to recurrence relations. The simplest cases are base-
expansions, and the standard Zeckendorf decomposition uses the Fibonacci
sequence. The expansions are finite sequences of nonnegative integer
coefficients (satisfying certain technical conditions to guarantee uniqueness
of the decomposition) and which can be viewed as analogs of sequences of
variable-length words made from some fixed alphabet. In this paper we present a
new approach and construction for uniform measures on expansions, identifying
them as the distribution of a Markov chain conditioned not to hit a set. This
gives a unified approach that allows us to easily recover results on the
expansions from analogous results for Markov chains, and in this paper we focus
on laws of large numbers, central limit theorems for sums of digits, and
statements on gaps (zeros) in expansions. We expect the approach to prove
useful in other similar contexts.Comment: Version 1.0, 25 pages. Keywords: Zeckendorf decompositions, positive
linear recurrence relations, distribution of gaps, longest gap, Markov
processe
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