23,301 research outputs found
Harris recurrence of Metropolis-within-Gibbs and trans-dimensional Markov chains
A -irreducible and aperiodic Markov chain with stationary probability
distribution will converge to its stationary distribution from almost all
starting points. The property of Harris recurrence allows us to replace
``almost all'' by ``all,'' which is potentially important when running Markov
chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms
are known to be Harris recurrent. In this paper, we consider conditions under
which Metropolis-within-Gibbs and trans-dimensional Markov chains are or are
not Harris recurrent. We present a simple but natural two-dimensional
counter-example showing how Harris recurrence can fail, and also a variety of
positive results which guarantee Harris recurrence. We also present some open
problems. We close with a discussion of the practical implications for MCMC
algorithms.Comment: Published at http://dx.doi.org/10.1214/105051606000000510 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Positive Harris recurrence and exponential ergodicity of the basic affine jump-diffusion
In this paper we find the transition densities of the basic affine
jump-diffusion (BAJD), which is introduced by Duffie and Garleanu [D. Duffie
and N. Garleanu, Risk and valuation of collateralized debt obligations,
Financial Analysts Journal 57(1) (2001), pp. 41--59] as an extension of the CIR
model with jumps. We prove the positive Harris recurrence and exponential
ergodicity of the BAJD. Furthermore we prove that the unique invariant
probability measure of the BAJD is absolutely continuous with respect to
the Lebesgue measure and we also derive a closed form formula for the density
function of .Comment: 21 pape
Affine Jump-Diffusions: Stochastic Stability and Limit Theorems
Affine jump-diffusions constitute a large class of continuous-time stochastic
models that are particularly popular in finance and economics due to their
analytical tractability. Methods for parameter estimation for such processes
require ergodicity in order establish consistency and asymptotic normality of
the associated estimators. In this paper, we develop stochastic stability
conditions for affine jump-diffusions, thereby providing the needed
large-sample theoretical support for estimating such processes. We establish
ergodicity for such models by imposing a `strong mean reversion' condition and
a mild condition on the distribution of the jumps, i.e. the finiteness of a
logarithmic moment. Exponential ergodicity holds if the jumps have a finite
moment of a positive order. In addition, we prove strong laws of large numbers
and functional central limit theorems for additive functionals for this class
of models
On some ergodicity properties for time inhomogeneous Markov processes with -periodic semigroup
We consider a time inhomogeneous strong Markov process
taking values in a Polish state space whose semigroup has a -periodic
structure. We give simple conditions which imply ergodicity of the grid chain
In case of -dimensional possibly degenerate
diffusions, the conditions are stated in terms of drift and diffusion
coefficient of the process
Ergodicity and limit theorems for degenerate diffusions with time periodic drift. Applications to a stochastic Hodgkin-Huxley model
We formulate simple criteria for positive Harris recurrence of strongly
degenerate stochastic differential equations with smooth coefficients when the
drift depends on time and space and is periodic in the time argument. There is
no time dependence in the diffusion coefficient. Our criteria rely on control
systems and the support theorem, existence of an attainable inner point of full
weak Hoermander dimension and of some Lyapunov function. Positive Harris
recurrence enables us to prove limit theorems for such diffusions.
As an application, we consider a stochastic Hodgkin-Huxley model for a
spiking neuron including its dendritic input. The latter carries some
deterministic periodic signal coded in its drift coefficient and is the only
source of noise for the whole system. This amounts to a 5d SDE driven by 1d
Brownian motion for which we can prove positive Harris recurrence using our
criteria. This approach provides us with laws of large numbers which allow to
describe the spiking activity of the neuron in the long run
Ergodicity for a stochastic Hodgkin-Huxley model driven by Ornstein-Uhlenbeck type input
We consider a model describing a neuron and the input it receives from its
dendritic tree when this input is a random perturbation of a periodic
deterministic signal, driven by an Ornstein-Uhlenbeck process. The neuron
itself is modeled by a variant of the classical Hodgkin-Huxley model. Using the
existence of an accessible point where the weak Hoermander condition holds and
the fact that the coefficients of the system are analytic, we show that the
system is non-degenerate. The existence of a Lyapunov function allows to deduce
the existence of (at most a finite number of) extremal invariant measures for
the process. As a consequence, the complexity of the system is drastically
reduced in comparison with the deterministic system.Comment: arXiv admin note: text overlap with arXiv:1207.019
Long-time behavior of stable-like processes
In this paper, we consider a long-time behavior of stable-like processes. A
stable-like process is a Feller process given by the symbol
where ,
and . More precisely, we give
sufficient conditions for recurrence, transience and ergodicity of stable-like
processes in terms of the stability function , the drift function
and the scaling function . Further, as a special case of
these results we give a new proof for the recurrence and transience property of
one-dimensional symmetric stable L\'{e}vy processes with the index of stability
Comment: To appear in: Stochastic Processes and their Application
MaxWeight Scheduling: Asymptotic Behavior of Unscaled Queue-Differentials in Heavy Traffic
The model is a "generalized switch", serving multiple traffic flows in
discrete time. The switch uses MaxWeight algorithm to make a service decision
(scheduling choice) at each time step, which determines the probability
distribution of the amount of service that will be provided. We are primarily
motivated by the following question: in the heavy traffic regime, when the
switch load approaches critical level, will the service processes provided to
each flow remain "smooth" (i.e., without large gaps in service)? Addressing
this question reduces to the analysis of the asymptotic behavior of the
unscaled queue-differential process in heavy traffic. We prove that the
stationary regime of this process converges to that of a positive recurrent
Markov chain, whose structure we explicitly describe. This in turn implies
asymptotic "smoothness" of the service processes.Comment: 9 page
Validity of heavy-traffic steady-state approximations in many-server queues with abandonment
We consider GI/Ph/n+M parallel-server systems with a renewal arrival process,
a phase-type service time distribution, n homogenous servers, and an
exponential patience time distribution with positive rate. We show that in the
Halfin-Whitt regime, the sequence of stationary distributions corresponding to
the normalized state processes is tight. As a consequence, we establish an
interchange of heavy traffic and steady state limits for GI/Ph/n+M queues
Linear Convergence on Positively Homogeneous Functions of a Comparison Based Step-Size Adaptive Randomized Search: the (1+1) ES with Generalized One-fifth Success Rule
In the context of unconstraint numerical optimization, this paper
investigates the global linear convergence of a simple probabilistic
derivative-free optimization algorithm (DFO). The algorithm samples a candidate
solution from a standard multivariate normal distribution scaled by a step-size
and centered in the current solution. This solution is accepted if it has a
better objective function value than the current one. Crucial to the algorithm
is the adaptation of the step-size that is done in order to maintain a certain
probability of success. The algorithm, already proposed in the 60's, is a
generalization of the well-known Rechenberg's Evolution Strategy (ES)
with one-fifth success rule which was also proposed by Devroye under the name
compound random search or by Schumer and Steiglitz under the name step-size
adaptive random search. In addition to be derivative-free, the algorithm is
function-value-free: it exploits the objective function only through
comparisons. It belongs to the class of comparison-based step-size adaptive
randomized search (CB-SARS). For the convergence analysis, we follow the
methodology developed in a companion paper for investigating linear convergence
of CB-SARS: by exploiting invariance properties of the algorithm, we turn the
study of global linear convergence on scaling-invariant functions into the
study of the stability of an underlying normalized Markov chain (MC). We hence
prove global linear convergence by studying the stability (irreducibility,
recurrence, positivity, geometric ergodicity) of the normalized MC associated
to the -ES. More precisely, we prove that starting from any initial
solution and any step-size, linear convergence with probability one and in
expectation occurs. Our proof holds on unimodal functions that are the
composite of strictly increasing functions by positively homogeneous functions
with degree (assumed also to be continuously differentiable). This
function class includes composite of norm functions but also non-quasi convex
functions. Because of the composition by a strictly increasing function, it
includes non continuous functions. We find that a sufficient condition for
global linear convergence is the step-size increase on linear functions, a
condition typically satisfied for standard parameter choices. While introduced
more than 40 years ago, we provide here the first proof of global linear
convergence for the -ES with generalized one-fifth success rule and the
first proof of linear convergence for a CB-SARS on such a class of functions
that includes non-quasi convex and non-continuous functions. Our proof also
holds on functions where linear convergence of some CB-SARS was previously
proven, namely convex-quadratic functions (including the well-know sphere
function)
- …