Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (02) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0,t] of the integral processwhereSis a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

Extreme values, range and weak convergence of integrals of Markov chains

1982 ◽  
Vol 19 (2) ◽  
pp. 272-288 ◽  
Author(s):  
P. J. Brockwell ◽  
S. I. Resnick ◽  
N. Pacheco-Santiago
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
State Space ◽  
Extreme Values ◽  
Finite State ◽  
Integral Process ◽  
Maximum Minimum ◽  
Finite State Space

A study is made of the maximum, minimum and range on [0, t] of the integral process where S is a finite state-space Markov chain. Approximate results are derived by establishing weak convergence of a sequence of such processes to a Wiener process. For a particular family of two-state stationary Markov chains we show that the corresponding centered integral processes exhibit the Hurst phenomenon to a remarkable degree in their pre-asymptotic behaviour.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (04) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (X t ) t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φ t ) t≥0 be an additive functional defined by φ t =∫0 t v(X s )d s. We consider the case in which the process (φ t ) t≥0 is oscillating and that in which (φ t ) t≥0 has a negative drift. In each of these cases, we condition the process (X t ,φ t ) t≥0 on the event that (φ t ) t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Antiduality and Möbius monotonicity: generalized coupon collector problem

2019 ◽  
Vol 23 ◽  
pp. 739-769
Author(s):  
Paweł Lorek
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Partial Ordering ◽  
Absorption Time ◽  
Absorbing Markov Chain ◽  
Coupon Collector ◽  
Strong Stationary Duality ◽  
Finite State ◽  
A Chain ◽  
Finite State Space

For a given absorbing Markov chain X* on a finite state space, a chain X is a sharp antidual of X* if the fastest strong stationary time (FSST) of X is equal, in distribution, to the absorption time of X*. In this paper, we show a systematic way of finding such an antidual based on some partial ordering of the state space. We use a theory of strong stationary duality developed recently for Möbius monotone Markov chains. We give several sharp antidual chains for Markov chain corresponding to a generalized coupon collector problem. As a consequence – utilizing known results on the limiting distribution of the absorption time – we indicate separation cutoffs (with their window sizes) in several chains. We also present a chain which (under some conditions) has a prescribed stationary distribution and its FSST is distributed as a prescribed mixture of sums of geometric random variables.

scholarly journals Sensitivity and convergence of uniformly ergodic Markov chains

2005 ◽  
Vol 42 (4) ◽  
pp. 1003-1014 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Rate Of Convergence ◽  
Transition Kernel ◽  
Numerical Examples ◽  
Perturbation Bounds ◽  
Finite State ◽  
New Perturbation ◽  
Finite State Space

For uniformly ergodic Markov chains, we obtain new perturbation bounds which relate the sensitivity of the chain under perturbation to its rate of convergence to stationarity. In particular, we derive sensitivity bounds in terms of the ergodicity coefficient of the iterated transition kernel, which improve upon the bounds obtained by other authors. We discuss convergence bounds that hold in the case of finite state space, and consider numerical examples to compare the accuracy of different perturbation bounds.

scholarly journals Non-homogeneous Markov chains with a finite state space and a Doeblin type theorem

1995 ◽  
Vol 18 (2) ◽  
pp. 365-370
Author(s):  
Rita Chattopadhyay
Keyword(s):  
Asymptotic Behavior ◽  
Markov Chains ◽  
State Space ◽  
Type Theorem ◽  
Nonnegative Matrices ◽  
Nonhomogeneous Markov Chains ◽  
Finite State ◽  
Finite State Space

Doeblin [1] considered some classes of finite state nonhomogeneous Markov chains and studied their asymptotic behavior. Later Cohn [2] considered another class of such Markov chains (not covered earlier) and obtained Doeblin type results. Though this paper does not present the “best possible” results, the method of proof will be of interest to the reader. It is elementary and based on Hajnal's results on products of nonnegative matrices.

scholarly journals Weak convergence of a numerical scheme for stochastic differential equations

2018 ◽  
Vol 37 (1) ◽  
pp. 201-215
Author(s):  
Esteban Aguilera ◽  
Raúl Fierro
Keyword(s):  
Differential Equation ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
State Space ◽  
Numerical Scheme ◽  
Step Size ◽  
Finite State ◽  
Convergence Of The Scheme ◽  
Finite State Space

WEAK CONVERGENCE OF A NUMERICAL SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONSIn this paper a numerical scheme approximating the solution to a stochastic differential equation is presented. On bounded subsets of time, this scheme has a finite state space, which allows us to decrease the round-off error when the algorithm is implemented. At the same time, the scheme introduced turns out locally consistent for any step size of time. Weak convergence of the scheme to the solution of the stochastic differential equation is shown.

Doubly stochastic Poisson processes and process control

1972 ◽  
Vol 4 (02) ◽  
pp. 318-338 ◽  
Author(s):  
Mats Rudemo
Keyword(s):  
Markov Chain ◽  
Differential Equations ◽  
State Space ◽  
Point Process ◽  
Doubly Stochastic ◽  
Long Run ◽  
Finite State ◽  
Control State ◽  
Special Case ◽  
Finite State Space

Consider a Poisson point process with an intensity parameter forming a Markov chain with continuous time and finite state space. A system of ordinary differential equations is derived for the conditional distribution of the Markov chain given observations of the point process. An estimate of the current intensity, optimal in the least-squares sense, is computed from this distribution. Applications to reliability and replacement theory are given. A special case with two states, corresponding to a process in control and out of control, is discussed at length. Adjustment rules, based on the conditional probability of the out of control state, are studied. Regarded as a function of time, this probability forms a Markov process with the unit interval as state space. For the distribution of this process, integro-differential equations are derived. They are used to compute the average long run cost of adjustment rules.

scholarly journals Sensitivity and convergence of uniformly ergodic Markov chains

2005 ◽  
Vol 42 (04) ◽  
pp. 1003-1014 ◽  
Author(s):  
A. Yu. Mitrophanov
Keyword(s):  
Markov Chains ◽  
State Space ◽  
Rate Of Convergence ◽  
Transition Kernel ◽  
Numerical Examples ◽  
Perturbation Bounds ◽  
Finite State ◽  
New Perturbation ◽  
Finite State Space

For uniformly ergodic Markov chains, we obtain new perturbation bounds which relate the sensitivity of the chain under perturbation to its rate of convergence to stationarity. In particular, we derive sensitivity bounds in terms of the ergodicity coefficient of the iterated transition kernel, which improve upon the bounds obtained by other authors. We discuss convergence bounds that hold in the case of finite state space, and consider numerical examples to compare the accuracy of different perturbation bounds.

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