16,774 research outputs found
Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors
It has been known since Ehrhard and Regnier's seminal work on the Taylor
expansion of -terms that this operation commutes with normalization:
the expansion of a -term is always normalizable and its normal form is
the expansion of the B\"ohm tree of the term. We generalize this result to the
non-uniform setting of the algebraic -calculus, i.e.
-calculus extended with linear combinations of terms. This requires us
to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's
techniques rely heavily on the uniform, deterministic nature of the ordinary
-calculus, and thus cannot be adapted; second is the absence of any
satisfactory generic extension of the notion of B\"ohm tree in presence of
quantitative non-determinism, which is reflected by the fact that the Taylor
expansion of an algebraic -term is not always normalizable. Our
solution is to provide a fine grained study of the dynamics of
-reduction under Taylor expansion, by introducing a notion of reduction
on resource vectors, i.e. infinite linear combinations of resource
-terms. The latter form the multilinear fragment of the differential
-calculus, and resource vectors are the target of the Taylor expansion
of -terms. We show the reduction of resource vectors contains the
image of any -reduction step, from which we deduce that Taylor expansion
and normalization commute on the nose. We moreover identify a class of
algebraic -terms, encompassing both normalizable algebraic
-terms and arbitrary ordinary -terms: the expansion of these
is always normalizable, which guides the definition of a generalization of
B\"ohm trees to this setting
Excursions of diffusion processes and continued fractions
It is well-known that the excursions of a one-dimensional diffusion process
can be studied by considering a certain Riccati equation associated with the
process. We show that, in many cases of interest, the Riccati equation can be
solved in terms of an infinite continued fraction. We examine the probabilistic
significance of the expansion. To illustrate our results, we discuss some
examples of diffusions in deterministic and in random environments.Comment: 28 pages. Minor changes to Section
A probabilistic interpretation of the parametrix method
In this article, we introduce the parametrix technique in order to construct
fundamental solutions as a general method based on semigroups and their
generators. This leads to a probabilistic interpretation of the parametrix
method that is amenable to Monte Carlo simulation. We consider the explicit
examples of continuous diffusions and jump driven stochastic differential
equations with H\"{o}lder continuous coefficients.Comment: Published at http://dx.doi.org/10.1214/14-AAP1068 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Kink estimation in stochastic regression with dependent errors and predictors
In this article we study the estimation of the location of jump points in the
first derivative (referred to as kinks) of a regression function \mu in two
random design models with different long-range dependent (LRD) structures. The
method is based on the zero-crossing technique and makes use of high-order
kernels. The rate of convergence of the estimator is contingent on the level of
dependence and the smoothness of the regression function \mu. In one of the
models, the convergence rate is the same as the minimax rate for kink
estimation in the fixed design scenario with i.i.d. errors which suggests that
the method is optimal in the minimax sense.Comment: 35 page
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