Expected hitting time markov chain

The expected first hitting time et is the expected first time for a stochastic variable to reach or cross a certain value. Antonina mitrofanova, nyu, department of computer science december 18, 2007 1 higher order transition probabilities very often we are interested in a probability of going from state i to state j in n steps, which we denote as pn ij. Next, lefebvre and guilbault and guilbault and lefebvre computed and, respectively, for a discretetime markov chain that tends to the ornsteinuhlenbeck process. A new approach to estimating the expected first hitting time. The expected hitting times for finite markov chains article. Hitting time and mixing time bounds of steins factors. We consider the problem of characterising expected hitting times and hitting probabilities for imprecise markov chains. The expected hitting times of markov chains as above, let p p ij i, j. Apr 24, 2018 find the expected time between light bulb replacements of the markov chain. Expected first hitting time in an absorbing markov chain. As another application, the same methodology applies to bound expected hitting time via steins factors. V denote the probability transition matrix of an irreducible ape riodic markov chain, and. An absorbing state is a state that, once entered, cannot be left.

In the absence of acceleration, the velocity formula gives distance travelled equals speed multiplied by time. Start from state 0 one of the partitions is empty and. Limiting conditional quasistationary distributions. This article highlights the interplay between steins method, modern markov chain theory and classical fluctuation theory. The graph on the right plots the probability mass in the lone absorbing state that represents the final square as the transition matrix is raised to larger and larger powers. To this end, we consider three distinct ways in which imprecise markov chains have been defined in the literature. Just as with discrete time, a continuous time stochastic process is a markov process if the conditional probability of a future event given the present state and additional information about past states depends only on the present state. Find the expected time between light bulb replacements of the markov chain. In continuous time, it is known as a markov process. Expected hitting time eht is one of the most critical concept in markov chains chen and zhang 2008. Learning outcomes by the end of this course, you should. For example, ifan algorithms execution time can be modelled as a hitting time in a markov chain, then average complexity bounds for the algorithm can readily be had.

Enter the transition matrix for problem 1 and run this script. It is named after the russian mathematician andrey markov. Is the hitting time for a continuous markov chain and a discrete markov chain the same. It is relatively easy to run a markov chain for a long time on a computer. Pdf hitting time, access time and optimal transport on graphs. If target forms a recurrent class, the elements of ht are expected absorption times. First hitting problems for markov chains that converge to a. The numbers next to the arrows are the transition probabilities. Apr 30, 2018 cs 70 markov chains hitting time, part 2 duration. To do this we consider the long term behaviour of such a markov chain. The expected hitting time of such chain has been calculated in chen and zhang, 2008, example 2. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields.

If n 100, and assuming each transition takes about 1400th of a second, the mean return time to an empty partition is 2100400 seconds. In this paper, using matrix algebra, we derive a new expression for the expected hitting times of a. A classic example of this is the ancient indian board game snakes and ladders. We conclude that a continuoustime markov chain is a special case of a semimarkov process. We also look at reducibility, transience, recurrence and periodicity. Homework 2 math 468568 solutions, spring 15 solution.

Naive asymptotics for hitting time bounds in markov chains. The state space of a markov chain, s, is the set of values that each x t can take. For example, if x t 6, we say the process is in state6 at timet. The authors also computed the quantity in the case when the markov chain is asymmetric as in lefebvre. The general idea of the method is to break down the possibilities resulting from the first step first transition in the markov chain. This and a sharper result are proved, and several related conjectures are discussed. The state space of a markov chain, s, is the set of values that each. In both cases, we use the law of total probability to derive a family of equations satisfied by the probabilitiestimes. For each class, mention if it is recurrent or transient. We saw that for an irreducible, aperiodic markov chain, the long time average of the fraction of time spent in a state i is. A great number of problems involving markov chains can be evaluated by a technique called first step analysis. Then xn is a markov chain we call this the smallworld markov chainwithstatespacesr. D is an arc set of ordered pairs of the elements of v, and i is a weight function on d, that is, to each i, j a d, we assign a.

Hitting times and probabilities for imprecise markov chains. June8,2012 abstract let 0 jun 17, 2014 unless the hitting probability is 1, the expected hitting time is infinite. As we will see in later section, a uniform continuous time markov chain can be constructed from a discrete time chain and an independent poisson process. Walds equation allows us to replace deterministic time kby the expected value of a random time. Google summer of code 2017 additions vandit jain august 2017 expected hitting time using ctmc the package provides expectedtime function to calculate average hitting time from one state to another. The correspondence between the terminologies of random walks and markov chains is given in table 5. An absorbing state is a state that is impossible to leave once reached.

The expected hitting times of markov chains as above, let p p ij i,jav denote the probability transition matrix of an irreducible ape riodic markov chain, and pi pi 1,pi 2. The expected hitting times for finite markov chains. It is named after the russian mathematician andrey markov markov chains have many applications as statistical models of realworld processes, such as studying cruise. First step analysis and fundamental matrix topics in.

Introduction let g v,d,ibe a weighted directed graph with loops, where v is a finite vertex set. Expected value and markov chains aquahouse tutoring. Moreover, we provide a characterisation of these quantities that directly generalises a similar characterisation for precise, homogeneous markov chains. The state of a markov chain at time t is the value ofx t.

A discrete time markov chain is a sequence of random variables x0. Games based entirely on chance can be modeled by an absorbing markov chain. This markov chain is irreducible because the process starting at any con guration, can reach any other con guration. Just as with discrete time, a continuoustime stochastic process is a markov process if the conditional probability of a future event given the present state and additional information about past states depends only on the present state. A state of a markov chain is persistent if it has the property that should the state ever be reached, the random process will return to it with probability one. Pdf given a discrete source distribution and discrete target distribution. A finite drunkards walk is an example of an absorbing markov chain. Then the expected time until they meet is bounded by a constant times the maximum first hitting time for the single chain.

The above description of a continuoustime stochastic process corresponds to a continuoustime markov chain. In the mathematical theory of probability, an absorbing markov chain is a markov chain in which every state can reach an absorbing state. Like general markov chains, there can be continuous time absorbing. Then use the law of total probability and markov property to derive a set. For a broad class of markov chains such as circulant markov chains or random walk on complete graphs, we prove a probabilistic analogue of the velocity formula between entropy and hitting time, where distance is the entropy of the markov trajectories from state i to state j in. Markov chains have many applications as statistical models. Expected value and markov chains karen ge september 16, 2016 abstract a markov chain is a random process that moves from one state to another such that the next state of the process depends only on where the process is at the present state. Velocity formulae between entropy and hitting time for markov. The expected hitting times for finite markov chains request pdf. A markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. I, where i is the set of all possible states and holding time qi 0 for every i 6 j. A discretetime markov chain is a kind of random process which involves dynamic systems represented as changing states along with time or steps. Giventhesources andtargett, thismarkovchainisparameterized by a and the function gt. Such a jump chain for 7 particles is displayed in fig.