Markov chains ergodic

When calculating the average copolymer composition and the probabilities P Uk] of the sequences of monomeric units it is possible to set Ta=0 in the expressions in (7), that is to neglect the finiteness of the size of the macromolecules. In this case the absorbing Markov chain (7) is replaced by the ergodic Markov chain with transition matrix Q whose elements ... 

If the n-steps transition probability elements are defined as the probability to reach the configuration j in n steps beginning from the configuration i and Ilj, = n (qjMarkov chain is ergodic (the ergodicity condition states that if i and j are two possible configurations with 0 and Ilj 0, for some finite n, pij(nl 0 ) and aperiodic (the chain of configurations do not form a sequence of events that repeats itself), the limits... 

This behavior is plausible recalling that the boundaries are of reflecting barrier type. In other words, the moving bird will never be at rest. A final remark is that the limiting behavior is independent of S(0), characterizing an ergodic Markov chain. 

A very interesting behavior may be obtained by varying the initial state vector S(0). It is observed that the steady state behavior is independent of S(0), where such a Markov chain, later discussed, is defined as ergodic. 

Results of the calculation of the state vectors S(n), are depicted in Fig.2-26 for different initial state vectors S(0) spelled out at the top of the figure. Thus for example, given that January 1st is a dry day (right-hand side of Fig.2-26), yields a probability of 0.580 that January 6th will be a dry. If January 1st is a wet day (middle of Fig.2-26), then the probability that January 6th is a dry day is 0.586. However, after ten days, the equilibrium conditions has for all practical purposes been reached. Thus, for example, if we call December 31st day 0 and January 10th day 10, then whatever distribution S[0] we take for day 0, we find that S(10) = [0.575, 0.425]. Such a Markov chain is defined as ergodic and is without memory to the initial state. 

Fig.45 shows results for two initial state vectors S(0). On the left hand-side the system is initially at Si, i.e., there are no customer in the queue. On the right hand-side the system is initially at S4, i.e., there are three customers in the queue. The propagation of the probability distribution indicates that the system (queue) reaches a steady state independent of S(0), i.e., an ergodic Markov chain. The calculation indicates that the steady state is achieved after nine steps and the appropriate state vector reads S(9) = [0.211, 0.212, 0.254, 0.322], i.e., S4 with three customers is the state of the highest probability. Certainly, if the possible number of states N (with maximum number of customers N -1) is changed, a new steady state would have been achieved, which depends also on the matrix, Eq.(2-93). 

Ergodic state. A finite Markov chain is ergodic if there exist probabilities Tik such that [6, p.41 4, p.lOl] ... 

As indicated before, the probability distribution ttk defined in Eqs.(2-106) and (2-107) is called a stationary distribution. If a Markov chain is ergodic, it can be shown [17, pp.247-255] that it possesses a unique stationary distribution that is, there exist Ttk that satisfy Eqs.(2-105), (2-106) and (2-107). There are Markov chains, however, that possess distributions that satisfy Eqs.(2-106) and (2-107), i.e., they have stationary distributions, which are not ergodic. For example, if the probability transition matrix is given by ... 

Some sufficient conditions for a finite Markov chain to be ergodic are based on the following theorems, given without proof [17, pp.247-255]. The first one states that A finite irreducible aperiodic Markov chain is ergodic. Let ... 

However, whether we have access to the configmrations in parallel or sequentially is irrelevant, which permits us to conclude that, for a sufficiently long and ergodic Markov chain, displacements will be accepted on average with the correct probability dictated by the principles of statistical physics (i.e., the probability density of a given statistical physical ensemble). 

A Markov chain is ergodic if it eventually reaches every state. If, in addition, a certain symmetry condition - the so-called criterion of detailed balance or microscopic reversibility - is fulfilled, the chain converges to the same stationary probability distribution of states, as we throw dice to decide which state transitions to take one after the other, no matter in which state we start. Thus, traversing the Markov chain affords us with an effective way of approximating its stationary probability distribution (Baldi Brunak, 1998). 

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... 

Once the limiting distribution is reached, it will persist this result is called the steady-state condition. If there exists one eigenvalue of unity, there exists only one limiting distribution, which is independent of the initial distribution. It can be shown that a unique limiting distribution exists if any state of the system can be reached from any other state with a nonzero multistep transition probability. In this case the system (or Markov chain) is ergodic. 

The term regular Markov chain, simply spoken, refers to the property that all possible states can be reached in a finite nnmber of snbsequent periods independent from the starting distribution. For more information abont definition, conditions, and properties of regular and ergodic Markov chains see e.g. Grinstead and SneU (1997, pp. 433 ff.). 

Because of the ergodic property of a Markov chain, the result at qr=jsds independent of the initial state of seeding, i.e.. the initial values X iq = 0), Xiy(q = 0). This means that the supramolecular approach generating polarity through a process of growth is circumventing the basic difficulty of creating polarity by spontaneous nucleation. 

The concept of ergodicity, and the method of analyzing it, can be understood using an analogy to a simpler finite model, a discrete Markov Chain. 

One approach to providing ergodicity to deterministic systems is to introduce random fluctuations via a Monte-Carlo technique [268]. Several Monte-Carlo methods are described in Appendix C. Randomized steps are taken and then an accept-reject mechanism is introduced in order to ensure that the steps are consistent with the canonical distribution. It is possible to combine the Metropolis-Hastings concept with timestepping procedures in a variety of ways, which are often subsumed under the title Monte-Carlo Markov Chain methods , these include... 

Under certain circumstances it can be shown that in the limit of large number of iterations of the process, the random variable X tends to X (in distribution). Thus the Monte-Carlo Markov Chain method can be used to generate sets of independent realizations of the random variable which can be used to calculate an expectation. In practice, one prefers to use a long sequence of the iterates from a single starting value. The efficacy (specifically the ergodicity and rate of convergence) of the Monte-Carlo Markov-Chain method depends on the choice of the transition density. 

Thus, we have seen that every binary DMS possesses the AEP in fact, so does every DMS. An information source need not be memoryless, though, to possess the AEP every stationary ergodic source—that is, a discrete source for which the output is a stationary ergodic random process—possesses the AEP (and, therefore, an entropy too). Even some nonstationary sources possess the AEP. For example, if the source sequence (wi, M2,. ..) is a time-invariant Markov chain, then its entropy is given by the formula... 

As with MMC it is essential that the RMC uses a Markov chain [1] for sampling states (so that the result is independent of the initial configuration) and that states are sampled ergodically. Assuming ergodicity this algorithm will result in fluctuations around the minimum of x -... 

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