Integral Limit Theorems for Nonlinear Boundary Crossing Time for Markov Chain

2013 ◽  
Vol 57 (1) ◽  
pp. 172-178
Author(s):  
F. Ragimov ◽  
F. Azizov
Keyword(s):  
Markov Chain ◽  
Limit Theorems ◽  
Nonlinear Boundary ◽  
Boundary Crossing ◽  
Integral Limit ◽  
Crossing Time

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (1) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain, obtained by restarting the original chain at a fixed state after each absorption. The limiting age, A(j), is the weak limit of the time given Xn = j (n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems for A (J) (J → ∞) are given for these examples.

scholarly journals Limit theorems for some continuous-time random walks

2011 ◽  
Vol 43 (3) ◽  
pp. 782-813 ◽  
Author(s):  
M. Jara ◽  
T. Komorowski
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Random Walks ◽  
Quantum Transport ◽  
Continuous Time ◽  
Limit Theorems ◽  
Transport Theory ◽  
Sufficient Conditions ◽  
Continuous Time Random Walks ◽  
Quantum Transport Theory

In this paper we consider the scaled limit of a continuous-time random walk (CTRW) based on a Markov chain {Xn,n≥ 0} and two observables, τ(∙) andV(∙), corresponding to the renewal times and jump sizes. Assuming that these observables belong to the domains of attraction of some stable laws, we give sufficient conditions on the chain that guarantee the existence of the scaled limits for CTRWs. An application of the results to a process that arises in quantum transport theory is provided. The results obtained in this paper generalize earlier results contained in Becker-Kern, Meerschaert and Scheffler (2004) and Meerschaert and Scheffler (2008), and the recent results of Henry and Straka (2011) and Jurlewicz, Kern, Meerschaert and Scheffler (2010), where {Xn,n≥ 0} is a sequence of independent and identically distributed random variables.

Boundary crossing for moving sums

1988 ◽  
Vol 25 (1) ◽  
pp. 81-88 ◽  
Author(s):  
J. Glaz ◽  
B. Johnson
Keyword(s):  
Standard Deviation ◽  
Random Variables ◽  
Boundary Crossing ◽  
Time Approximation ◽  
Crossing Time ◽  
Moving Sum

Let Xi, i ≧ 1, be a sequence of independent N(0, 1) random variables and Sj,m = Xj + · ·· + Xj+m–1, the jth moving sum. Let τ m = inf{j ≧ 1 : Sj,m > a} + m – 1, the boundary crossing time. Approximation in the spirit of Glaz and Johnson (1984), (1986) and Samuel-Cahn (1983) are given for Pr(τm > n), E(τ m), and σ (τ m),the standard deviation of τm.

On the age distribution of a Markov chain

1978 ◽  
Vol 15 (01) ◽  
pp. 65-77 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Limit Theorems ◽  
Age Distribution ◽  
Weak Limit ◽  
Absorbing State ◽  
Absorbing Markov Chain ◽  
Continuous Random Walk ◽  
A Chain ◽  
Watson Process

This paper develops the notion of the limiting age of an absorbing Markov chain, conditional on the present state. Chains with a single absorbing state {0} are considered and with such a chain can be associated a return chain,obtained by restarting the original chain at a fixed state after each absorption. The limiting age,A(j), is the weak limit of the timegivenXn=j(n → ∞).A criterion for the existence of this limit is given and this is shown to be fulfilled in the case of the return chains constructed from the Galton–Watson process and the left-continuous random walk. Limit theorems forA(J) (J →∞) are given for these examples.

Some integral limit theorems for additive functions

1977 ◽  
Vol 16 (4) ◽  
pp. 564-573
Author(s):  
B. V. Levin ◽  
N. M. Timofeev
Keyword(s):  
Limit Theorems ◽  
Additive Functions ◽  
Integral Limit

Central limit theorems for multivariate semi-Markov sequences and processes, with applications

1999 ◽  
Vol 36 (2) ◽  
pp. 415-432 ◽  
Author(s):  
Frank Ball
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Central Limit ◽  
Limit Theorems ◽  
Markov Models ◽  
Central Limit Theorems ◽  
Renewal Processes ◽  
Counting Processes ◽  
Markov Renewal Processes ◽  
Semi Markov Processes

In this paper, central limit theorems for multivariate semi-Markov sequences and processes are obtained, both as the number of jumps of the associated Markov chain tends to infinity and, if appropriate, as the time for which the process has been running tends to infinity. The theorems are widely applicable since many functions defined on Markov or semi-Markov processes can be analysed by exploiting appropriate embedded multivariate semi-Markov sequences. An application to a problem in ion channel modelling is described in detail. Other applications, including to multivariate stationary reward processes, counting processes associated with Markov renewal processes, the interpretation of Markov chain Monte Carlo runs and statistical inference on semi-Markov models are briefly outlined.

scholarly journals DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

2019 ◽  
Vol 34 (2) ◽  
pp. 235-257
Author(s):  
Peter Spreij ◽  
Jaap Storm
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Regime Switching ◽  
Limit Theorems ◽  
Counting Process ◽  
Limit Behavior ◽  
Diffusion Approximations ◽  
Default Rate ◽  
Markov Modulated ◽  
Rapid Switching

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

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