MULTIPLICATIVE STOCHASTIC PERTURBATIONS OF ONE-DIMENSIONAL MAPS

2013 ◽  
Vol 13 (02) ◽  
pp. 1250020 ◽  
Author(s):  
YUKIKO IWATA
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Noise Level ◽  
Random Perturbation ◽  
Invariant Probability Measure ◽  
Random Perturbations ◽  
Stochastic Perturbations ◽  
One Dimensional ◽  
One Dimensional Maps

We consider random perturbations of some one-dimensional map S : [0, 1] → [0, 1] such that [Formula: see text] parametrized by 0 < ε < 1, where {Cn} is an i.i.d. sequence. We prove that this random perturbation is small with respect to the noise level 0 < ε < 1 and give a class of one-dimensional maps for which there always exists a smooth invariant probability measure for the Markov process {Xn}n≥0.

scholarly journals STATISTICAL PROPERTIES OF ONE-DIMENSIONAL MAPS WITH CRITICAL POINTS AND SINGULARITIES

2006 ◽  
Vol 06 (04) ◽  
pp. 423-458 ◽  
Author(s):  
K. DÍAZ-ORDAZ ◽  
M. P. HOLLAND ◽  
S. LUZZATTO
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Invariant Probability Measure ◽  
Return Time ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Time Asymptotics ◽  
Exponential Decay Of Correlations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

We prove that a class of one-dimensional maps with an arbitrary number of non-degenerate critical and singular points admits an induced Markov tower with exponential return time asymptotics. In particular the map has an absolutely continuous invariant probability measure with exponential decay of correlations for Hölder observations.

ON DYNAMICAL MODEL OF DRILLING WELLS

1999 ◽  
Vol 09 (09) ◽  
pp. 1705-1718 ◽  
Author(s):  
G. CHAKVETADZE ◽  
A. STEPIN
Keyword(s):  
Continuous Dependence ◽  
Dimensionless Parameter ◽  
Invariant Probability Measure ◽  
Sufficient Condition ◽  
Weak Mixing ◽  
Absolutely Continuous ◽  
Stochastic Perturbations ◽  
One Dimensional ◽  
Absolutely Continuous Invariant ◽  
Mixing Property

A family of one-dimensional mappings and their stochastic perturbations, related to the determining of cogged bits efficiency, is studied. A sufficient condition for the existence of absolutely continuous invariant probability measure is given, and the weak mixing property of this measure is established. We formulate and discuss the results concerning the stochastic stability of the invariant measure and its continuous dependence on the dimensionless parameter of the model. Some new problems are also outlined.

scholarly journals Convergence in distribution of the multi-dimensional Kohonen algorithm

2001 ◽  
Vol 38 (1) ◽  
pp. 136-151 ◽  
Author(s):  
Ali A. Sadeghi
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Topological Space ◽  
Wide Class ◽  
Invariant Probability Measure ◽  
Learning Rate ◽  
Important Case ◽  
Convergence In Distribution ◽  
Special Shape ◽  
Data Space

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

1992 ◽  
Vol 12 (1) ◽  
pp. 13-37 ◽  
Author(s):  
Michael Benedicks ◽  
Lai-Sang Young
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Random Perturbations ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Small Random Perturbations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

AbstractWe study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (1) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

Convergence in Distribution of the One-Dimensional Kohonen Algorithms when the Stimuli are not Uniform

1994 ◽  
Vol 26 (01) ◽  
pp. 80-103 ◽  
Author(s):  
Catherine Bouton ◽  
Gilles Pagès
Keyword(s):  
Markov Chain ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Absolute Continuity ◽  
Invariant Probability Measure ◽  
Convergence In Distribution ◽  
One Dimensional ◽  
Open Set ◽  
Recurrent Markov Chain ◽  
The One

We show that the one-dimensional self-organizing Kohonen algorithm (with zero or two neighbours and constant step ε) is a Doeblin recurrent Markov chain provided that the stimuli distribution μ is lower bounded by the Lebesgue measure on some open set. Some properties of the invariant probability measure vε (support, absolute continuity, etc.) are established as well as its asymptotic behaviour as ε ↓ 0 and its robustness with respect to μ.

scholarly journals Ergodic properties and ergodic decompositions of continuous-time Markov processes

2006 ◽  
Vol 43 (3) ◽  
pp. 767-781 ◽  
Author(s):  
O. L. V. Costa ◽  
F. Dufour
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Continuous Time ◽  
Invariant Probability Measure ◽  
Probability Measures ◽  
Transition Kernel ◽  
Occupation Measure ◽  
Ergodic Properties ◽  
Necessary And Sufficient ◽  
Separable Metric Space

In this paper we obtain some ergodic properties and ergodic decompositions of a continuous-time, Borel right Markov process taking values in a locally compact and separable metric space. Initially, we assume that an invariant probability measure (IPM) μ exists for the process and, without making any further assumptions on the transition kernel, obtain some characterization results for the convergence of the expected occupation measure to a limit kernel. Under the same assumption, we present the so-called Yosida decomposition. Next, instead of assuming the existence of an IPM, we assume that the Markov process satisfies a certain condition, named the T'-condition. Under this condition it is shown that the Foster-Lyapunov criterion is necessary and sufficient for the existence of an IPM and that the process admits a Doeblin decomposition. Furthermore, it is shown that in this case the set of ergodic probability measures is countable and that every probability measure for the Markov process is nonsingular with respect to the transition kernel.

scholarly journals Adapted random perturbations for non-uniformly expanding maps

2014 ◽  
Vol 14 (04) ◽  
pp. 1450007
Author(s):  
Vitor Araujo ◽  
Maria Jose Pacifico ◽  
Mariana Pinheiro
Keyword(s):  
Critical Points ◽  
Stochastic Stability ◽  
Random Perturbation ◽  
Random Perturbations ◽  
Noise Levels ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Time Map ◽  
Uniform Expansion ◽  
Hyperbolic Times

We obtain stochastic stability of C2 non-uniformly expanding one-dimensional endomorphisms, requiring only that the first hyperbolic time map be Lp-integrable for p > 3. We show that, under this condition (which depends only on the unperturbed dynamics), we can construct a random perturbation that preserves the original hyperbolic times of the unperturbed map and, therefore, to obtain non-uniform expansion for random orbits. This ensures that the first hyperbolic time map is uniformly integrable for all small enough noise levels, which is known to imply stochastic stability. The method enables us to obtain stochastic stability for a class of maps with infinitely many critical points. For higher dimensional endomorphisms, a similar result is obtained, but under stronger assumptions.

scholarly journals Convergence in distribution of the multi-dimensional Kohonen algorithm

2001 ◽  
Vol 38 (01) ◽  
pp. 136-151
Author(s):  
Ali A. Sadeghi
Keyword(s):  
Markov Process ◽  
Probability Measure ◽  
Topological Space ◽  
Wide Class ◽  
Invariant Probability Measure ◽  
Learning Rate ◽  
Important Case ◽  
Convergence In Distribution ◽  
Special Shape ◽  
Data Space

Here we consider the Kohonen algorithm with a constant learning rate as a Markov process evolving in a topological space. Despite the fact that the algorithm is not weak Feller, we show that it is a T-chain, regardless of the dimensionalities of both data space and network and the special shape of the neighborhood function. In addition for the practically important case of the multi-dimensional setting, it is shown that the chain is irreducible and aperiodic. We show that these imply the validity of Doeblin's condition, which in turn ensures the convergence in distribution of the process to an invariant probability measure with a geometric rate. Furthermore, it is shown that the process is positive Harris recurrent, which enables us to use statistical devices to measure the centrality and variability of the invariant probability measure. Our results cover a wide class of neighborhood functions.

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