28,969 research outputs found
Finiteness of hitting times under taboo
We consider a continuous-time Markov chain with a finite or countable state
space. For a site y and subset H of the state space, the hitting time of y
under taboo H is defined to be infinite if the process trajectory hits H before
y, and the first hitting time of y otherwise. We investigate the probability
that such times are finite. In particular, if the taboo set is finite, an
efficient iterative scheme reduces the study to the known case of a singleton
taboo. A similar procedure applies in the case of finite complement of the
taboo set. The study is motivated by classification of branching processes with
finitely many catalysts.
Keywords and phrases: Markov chain, hitting time, taboo probabilities,
catalytic branching process
Strong and weak convergence of population size in supercritical catalytic branching process
A general model of catalytic branching process (CBP) with any finite number
of catalysis centers in a discrete space is studied. More exactly, it is
assumed that particles move in this space according to a specified Markov chain
and they may produce offspring only in the presence of catalysts located at
fixed points. The asymptotic (in time) behavior of the total number of
particles as well as the local particles numbers is investigated. The problems
of finding the global extinction probability and local extinction probability
are solved. Necessary and sufficient conditions are established for phase of
pure global survival and strong local survival. Under wide conditions the limit
theorems for the normalized total and local particles numbers in supercritical
CBP are proved in the sense of almost surely convergence as well as with
respect to convergence in distribution. Generalizations of a number of previous
results are obtained as well. In the proofs the main role is played by recent
results by the author devoted to classification of CBP and the moment analysis
of the total and local particles numbers in CBP.
Keywords and phrases: catalytic branching process, extinction probability,
strong local survival, pure global phase, limit theorems, multi-type
Bellman-Harris process
Catalytic Branching Random Walk with Semi-exponential Increments
A catalytic branching random walk on a multidimensional lattice, with
arbitrary finite number of catalysts, is studied in supercritical regime. The
dynamics of spatial spread of the particles population is examined, upon
normalization. The components of the vector random walk jump are assumed
independent (or close to independent) and have semi-exponential distributions
with, possibly, different parameters. A limit theorem on the almost sure
normalized positions of the particles at the population ``front'' is
established. Contrary to the case of the random walk increments with ``light''
distribution tails, studied by Carmona and Hu (2014) in one-dimensional setting
and Bulinskaya (2018) in multidimensional setting, the normalizing factor has a
power rate and grows faster than linear in time function. The limiting shape of
the front in the case of semi-exponential tails is non-convex in contrast to a
convex one in the case of light tails.Comment: 1 figur
Maximum of Catalytic Branching Random Walk with Regularly Varying Tails
For a continuous-time catalytic branching random walk (CBRW) on Z, with an
arbitrary finite number of catalysts, we study the asymptotic behavior of
position of the rightmost particle when time tends to infinity. The mild
requirements include the regular variation of the jump distribution tail for
underlying random walk and the well-known L log L condition for the offspring
numbers. In our classification, given in the previous paper, the analysis
refers to supercritical CBRW. The principle result demonstrates that, after a
proper normalization, the maximum of CBRW converges in distribution to a
non-trivial law. An explicit formula is provided for this normalization and
non-linear integral equations are obtained to determine the limiting
distribution function. The novelty consists in establishing the weak
convergence for CBRW with "heavy" tails, in contrast to the known behavior in
case of "light" tails of the random walk jumps. The new tools such as
"many-to-few lemma" and spinal decomposition appear non-efficient here. The
approach developed in the paper combines the techniques of renewal theory,
Laplace transform, non-linear integral equations and large deviations theory
for random sums of random variables.
Keywords and phrases: catalytic branching random walk, heavy tails, regular
varying tails, spread of population, L log L condition
Correlation Functions of Local Operators in 2D Gravity Coupled to Minimal Matter
Recent advances are being discussed on the calculation, within the conformal
field theory approach, of the correlation functions for local operators in the
theory of 2D gravity coupled to the minimal models of matter.Comment: 12 page
Naive model from 1970th applied to CMR manganites: it seems to work
Existing experimental data for various colossal magnetoresistance manganites
have been examined employing an ovesimplified model that roots in 1970th. This
model considers a classical semiconductor where conducting bands are affected
by the strong Weiss exchange field that arises from the magnetic order in the
substance. The field--caused shifts of the conducting bands results in the
change in the number of thermally activated carriers, and this change is
presumed to be responsible for the resistivity dependences on temperature and
magnetic field and for the CMR itself. Employing this model we calculate this
hypothetical Weiss field from the experimental data for various CMR manganites
employing minimal set of the adjustable parameters, namely two. The obtained
Weiss field behaves with temperature and external field similarly to the local
magnetization, its supposed source, hence supporting the model.Comment: 14 pages, 7 figure
Classification of catalytic branching processes and structure of the criticality set
We study a catalytic branching process (CBP) with any finite set of
catalysts. This model describes a system of particles where the movement is
governed by a Markov chain with arbitrary finite or countable state space and
the branching may only occur at the points of catalysis. The results obtained
generalize and strengthen those known in cases of CBP with a single catalyst
and of branching random walk on d-dimensional integer lattice with a finite
number of sources of particles generation. We propose to classify CBP with N
catalysts as supercritical, critical or subcritical according to the value of
the Perron root of a specified NxN matrix. Such classification agrees with the
moment analysis performed here for local and total particles numbers. By
introducing the criticality set C we also consider the influence of catalysts
parameters on the process behavior. The proof is based on construction of
auxiliary multi-type Bellman-Harris processes with the help of hitting times
under taboo and on application of multidimensional renewal theorems.
Keywords and phrases: catalytic branching process, classification, hitting
times under taboo, moment analysis, multi-type Bellman-Harris process.Comment: 1 figur
Spread of a Catalytic Branching Random Walk on a Multidimensional Lattice
For a supercritical catalytic branching random walk on Z^d (d is positive
integer) with an arbitrary finite catalysts set we study the spread of
particles population as time grows to infinity. Namely, we divide by t the
position coordinates of each particle existing at time t and then let t tend to
infinity. It is shown that in the limit there are a.s. no particles outside the
closed convex surface in R^d which we call the propagation front and, under
condition of infinite number of visits of the catalysts set, a.s. there exist
particles on the propagation front. We also demonstrate that the propagation
front is asymptotically densely populated and derive its alternative
representation. Recent strong limit theorems for total and local particles
numbers established by the author play an essential role. The results obtained
develop ones by Ph.Carmona and Y.Hu (2014) devoted to the spread of catalytic
branching random walk on Z.
Keywords and phrases: branching random walk, supercritical regime, spread of
population, propagation front, many-to-one lemma.Comment: 2 figure
Subcritical catalytic branching random walk with finite or infinite variance of offspring number
Subcritical catalytic branching random walk on d-dimensional lattice is
studied. New theorems concerning the asymptotic behavior of distributions of
local particles numbers are established. To prove the results different
approaches are used including the connection between fractional moments of
random variables and fractional derivatives of their Laplace transforms. In the
previous papers on this subject only supercritical and critical regimes were
investigated assuming finiteness of the first moment of offspring number and
finiteness of the variance of offspring number, respectively. In the present
paper for the offspring number in subcritical regime finiteness of the moment
of order 1+\delta is required where \delta is some positive number.
Keywords and phrases: branching random walk, subcritical regime, finite
variance of offspring number, infinite variance of offspring number, local
particles numbers, limit theorems, fractional moments, fractional derivatives
Online Handwritten Devanagari Stroke Recognition Using Extended Directional Features
This paper describes a new feature set, called the extended directional
features (EDF) for use in the recognition of online handwritten strokes. We use
EDF specifically to recognize strokes that form a basis for producing
Devanagari script, which is the most widely used Indian language script. It
should be noted that stroke recognition in handwritten script is equivalent to
phoneme recognition in speech signals and is generally very poor and of the
order of 20% for singing voice. Experiments are conducted for the automatic
recognition of isolated handwritten strokes. Initially we describe the proposed
feature set, namely EDF and then show how this feature can be effectively
utilized for writer independent script recognition through stroke recognition.
Experimental results show that the extended directional feature set performs
well with about 65+% stroke level recognition accuracy for writer independent
data set.Comment: 8th International Conference on Signal Processing and Communication
Systems 15 - 17 December 2014, Gold Coast, Australi
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