Finite Markov processes

Here, / is the outcome function, the value of which is Sj if the outcome of the th step is sj. A finite Markov process is a finite Markov chain if the transition probabilities do not depend on n. Then, the transition matrix for a Markov chain is the matrix P with entries py. 

For finite Markov processes, it can be proved that for any finite Markov chain, no matter where the walker starts, the probability that the walker is in an ergodic state after n steps tends to unity as n oc. Thus, powers of Q in the above aggregated version of P tend to 0 and consequently for any absorbing Markov chain, the matrix I - Q has an inverse N, called the fundamental matrix. In the problem defined by Eq. (4.12) the matrix N is... 

M. losifescu. Finite Markov Processes and Their Applications (Wiley, Chichester, 1980). 

Equation (3-325), along with the fact that Y(t) has zero mean and is gaussian, completely specifies Y(t) as a random process. Detailed expressions for the characteristic function of the finite order distributions of Y(t) can be calculated by means of Eq. (3-271). A straightforward, although somewhat tedious, calculation of the characteristic function of the finite-order distributions of the gaussian Markov process defined by Eq. (3-218) now shows that these two processes are in fact identical, thus proving our assertion. 

The compulsory fulfillment of conditions (4.2) and (4.3) physically follows from the fact that a one-dimensional Markov process is nondifferentiable that is, the derivative of Markov process has an infinite variance (instantaneous speed is an infinitely high). However, the particle with the probability equals unity drifts for the finite time to the finite distance. That is why the particle velocity changes its sign during the time, and the motion occurs in an opposite directions. If the particle is located at some finite distance from the boundary, it cannot reach the boundary in a trice—the condition (4.2). On the contrary, if the particle is located near a boundary, then it necessarily crosses the boundary— the condition (4.3). 

A finite Markov chain is one whose range consists of a finite number N of states. They have been extensively studied, because they are the simplest Markov processes that still exhibit most of the relevant features. The first probability distribution Pi(y, t) is an iV-component vector pn(t) (n = 1,2,...,JV). The transition probability Tz(y2 yi) is an N x N matrix. The Markov property (3.3) leads to the matrix equation... 

Remark. Consider a Markov process that can be visualized as a particle jumping back and forth among a finite number of sites m, with constant probabilities per unit time. Suppose it has a single stationary distribution psn, with the property (5.3). After an initial period it will be true that, if I pick an arbitrary t, the probability to find the particle at n is ps . That implies that psn is the fraction of its life that the particle spends at site n, once equilibrium has been reached. This fact is called ergodicity. For a Markov process with finitely many sites ergodicity is tantamount to indecom-posability. ) In (VII.7.13) a more general result for the times spent at the various sites is obtained. 

We begin the discussion by referring to the stochastic model given by relation (4.58), which is rewritten here as shown in relation (4.120). Here for a finite Markov connection process we must consider the constant time values for all the elements of the matrix P = pij i k-... 

The characteristics of the state space being measured can be used to classify the Markov process. For most purposes, a discrete or finite space is assumed and this implies that there are a finite number of states that will be reached by the process (14). A continuous or infinite process is also possible. Time intervals of observation of a process can be used to classify a Markov process. Processes can be observed at discrete or restricted intervals, or continuously (15). 

A (first-order) Markov process is defined as a finite-state probability model in which only the current state and the probability of each possible state change is known. Thus, the probability of making a transition to each state of the process, and thus the trajectory of states in the future, depends only upon the current state. A Markov process can be used to model random but dependent events. Given the observations from a sequence of events (Markov chain), one can determine the probability of one element of the chain (state) being followed by another, thus constructing a stochastic model of the system being observed. For instance, a first-order Markov chain can be defined as... 

Two strategies that can be used to simplify the calculation of the mean walklength n) are now reviewed. In the first of these, it is shown that if the random walk is modeled by a stationary Markov process on a finite state... 

The study of the development of these profiles over time by using a probabilistic graph of transitions between the clusters inferred by k-TSSI (k-Testable Languages in the Strict Sense Inference) algorithm. The objective is to deduce Markov process which has a discrete (finite or countable) state-space. 

The follow-up of reactions or with A, respectively, A + or B " with B is a chain process known as a finite Markov chain. This is in accordance with the kink site property that the last atom added to the kink site position determines the nature and activity of the kink site. Applications to deposition processes were published. " ... 

Consider a n-components dynamic system described by an irreducible homogenous Markov process = Xj, t > 0 (initial state /) with finite state space E and the transition rate matrix M. This Markov process is ergodic and a single stationary distribution exists (Ross 1996). Let a row vector jt = (tti, 7T2,. ..) be the vector of steady state probabilities (stationary distribution vector). Chapman-Kolmogorov equations at steady state can be written as ... 

The time to stopping in a state P is the same as the first passage time Tk for this state. It follows from the fact that P is absorbing. Consider an finite irreducible Markov process Xft), i > 0 with states in S U Ek]-Assume that all states in this process are transient but Ek which is absorbing. The transient intensity matrix of this process can be expressed as... 

Definition A continuous distribution H(-) on [0,oo[ is a phase-type one with representation (o ,T) if it is the distribution of the time imtU the absorption in a finite state Markov process with one ahsorhent state. Denoting by a the initial probabUity vector of the Markov process, the generator is given by... 

In contrast to the continuous models, the discrete models consider the processes at the level of individual structural elements, e.g. individual fibres, threads or loops, or individual stages of the process. In these models the processes are modelled as a series of states where the transition from one state to another happens with a probability. The underpinning theories for these models are theory of Markov processes (Kemeny and Snell, 1960), queuing theory (Gross et nf, 2008), and finite automata theory (Anderson, 2006 Hopcroft et al., 2007). 

A further extension of these ideas, in which multiple states that evolve in time are possible, is obtained when one models the speech signal by a hidden Markov process (HMP) [8]. An HMP is a bivariate random process of states and observations sequences. The state process S t = 1,2,... is a finite-state homogeneous Markov chain that is not directly observed. The observation process yf,t = 1,2,...) is conditionally independent given the state process. Thus, each observation depends statistically only on the state of the Markov chain at the same time and not on any other states or observations. Consider, for example, an HMP observed in an additive white noise process W),t = 1,2,...). For each t, let Zt = Yt + Wt denote the noisy signal. Let Z = Zi,..., Z,. Let / denote the number of states of the Markov chain. The causal MMSE estimator of Y, given Z is given by [6]... 

---