On Sieved Orthogonal Polynomials II: Random Walk Polynomials

1986 ◽  
Vol 38 (2) ◽  
pp. 397-415 ◽  
Author(s):  
Jairo Charris ◽  
Mourad E. H. Ismail
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Orthogonal Polynomials ◽  
Transition Probabilities ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Markov Process ◽  
Image Position

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities(1.1)satisfy(1.2)as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

A linear birth and death process under the influence of another process

1975 ◽  
Vol 12 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Markov Process ◽  
Closed Form ◽  
Limit Distribution ◽  
Explicit Solution ◽  
Negative Integer ◽  
Death Process ◽  
Birth And Death Process ◽  
Death Rates ◽  
Time To Extinction

Let {X1 (t), X2 (t), t ≧ 0} be a bivariate birth and death (Markov) process taking non-negative integer values, such that the process {X2(t), t ≧ 0} may influence the growth of the process {X1(t), t ≧ 0}, while the process X2 (·) itself grows without any influence whatsoever of the first process. The process X2 (·) is taken to be a simple linear birth and death process with λ2 and µ2 as its birth and death rates respectively. The process X1 (·) is also assumed to be a linear birth and death process but with its birth and death rates depending on X2 (·) in the following manner: λ (t) = λ1 (θ + X2 (t)); µ(t) = µ1 (θ + X2 (t)). Here λ i, µi and θ are all non-negative constants. By studying the process X1 (·), first conditionally given a realization of the process {X2 (t), t ≧ 0} and then by unconditioning it later on by taking expectation over the process {X2 (t), t ≧ 0} we obtain explicit solution for G in closed form. Again, it is shown that a proper limit distribution of X1 (t) always exists as t→∞, except only when both λ1 > µ1 and λ2 > µ2. Also, certain problems concerning moments of the process, regression of X1 (t) on X2 (t); time to extinction, and the duration of the interaction between the two processes, etc., are studied in some detail.

scholarly journals A Diffusion Limit for Generalized Correlated Random Walks

2006 ◽  
Vol 43 (01) ◽  
pp. 60-73 ◽  
Author(s):  
Urs Gruber ◽  
Martin Schweizer
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Transition Probabilities ◽  
Partial Sums ◽  
Diffusion Limit ◽  
Correlated Random Walk ◽  
Binomial Trees ◽  
Correlated Random Walks ◽  
Limiting Behaviour ◽  
Image Position

A generalized correlated random walk is a process of partial sums such that (X, Y) forms a Markov chain. For a sequence (X n ) of such processes in which each takes only two values, we prove weak convergence to a diffusion process whose generator is explicitly described in terms of the limiting behaviour of the transition probabilities for the Y n . Applications include asymptotics of option replication under transaction costs and approximation of a given diffusion by regular recombining binomial trees.

scholarly journals Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

2015 ◽  
Vol 52 (1) ◽  
pp. 278-289 ◽  
Author(s):  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Symmetric Matrix ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Absorbing State ◽  
State 1 ◽  
Birth Death

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

scholarly journals On selecting the most reliable components

1998 ◽  
Vol 2 (2) ◽  
pp. 133-145 ◽  
Author(s):  
Dylan Shi
Keyword(s):  
Markov Process ◽  
Death Process ◽  
Series System ◽  
Birth And Death Process ◽  
Retrograde Motion ◽  
The Past ◽  
Optimal Policies ◽  
Average Proportion ◽  
The Mean ◽  
Mean Time

Consider a series system consisting of n components of k types. Whenever a unit fails, it is replaced immediately by a new one to keep the system working. Under the assumption that all the life lengths of the components are independent and exponentially distributed and that the replacement policies depend only on the present state of the system at each failure, the system may be represented by a birth and death process. The existence of the optimum replacement policies are discussed and the ε-optimal policies axe derived. If the past experience of the system can also be utilized, the process is not a Markov process. The optimum Bayesian policies are derived and the properties of the resulting process axe studied. Also, the stochastic processes are simulated and the probability of absorption, the mean time to absorption and the average proportion of the retrograde motion are approximated.

On the differential equations for the transition probabilities of Markov processes with enumerably many states

1953 ◽  
Vol 49 (2) ◽  
pp. 247-262 ◽  
Author(s):  
G. E. H. Reuter ◽  
W. Ledermann ◽  
M. S. Bartlett
Keyword(s):  
Markov Process ◽  
Differential Equations ◽  
Conditional Probability ◽  
Markov Processes ◽  
Transition Probabilities ◽  
Regularity Conditions ◽  
Kolmogorov Equations ◽  
Image Position ◽  
Positive Integers

Let pik (s, t) (i, k = 1, 2, …; s ≤ t) be the transition probabilities of a Markov process in a system with an enumerable set of states. The states are labelled by positive integers, and pik (s, t) is the conditional probability that the system be in state k at time t, given that it was in state i at an earlier time s. If certain regularity conditions are imposed on the pik, they can be shown to satisfy the well-known Kolmogorov equations§

scholarly journals Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

2015 ◽  
Vol 52 (01) ◽  
pp. 278-289 ◽  
Author(s):  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Symmetric Matrix ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Absorbing State ◽  
State 1 ◽  
Birth Death

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

Level phase independence for GI/M/1-type Markov chains

2000 ◽  
Vol 37 (4) ◽  
pp. 984-998 ◽  
Author(s):  
Guy Latouche ◽  
P. G. Taylor
Keyword(s):  
Phase Transitions ◽  
Markov Chains ◽  
State Space ◽  
Transition Probabilities ◽  
The Other ◽  
One Dimension ◽  
Death Process ◽  
Two Dimensional ◽  
Birth And Death Process ◽  
Special Case

GI/M/1-type Markov chains make up a class of two-dimensional Markov chains. One dimension is usually called the level, and the other is often called the phase. Transitions from states in level k are restricted to states in levels less than or equal to k+1. For given transition probabilities in the interior of the state space, we show that it is always possible to define the boundary transition probabilities in such a way that the level and phase are independent under the stationary distribution. We motivate our analysis by first considering the quasi-birth-and-death process special case in which transitions from any state are restricted to states in the same, or adjacent, levels.

A continuous-time analogue of random walk in a random environment

1980 ◽  
Vol 17 (1) ◽  
pp. 259-264 ◽  
Author(s):  
Grant Ritter
Keyword(s):  
Random Walk ◽  
Random Environment ◽  
Continuous Time ◽  
Random Variables ◽  
Death Process ◽  
Birth And Death Process ◽  
Time Parameters

By making the time parameters of a birth and death process random variables, we create a continuous-time analogue of random walk in a random environment. Criteria for recurrence or transience are discussed and an a.s. convergence law is determined.

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