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A classic example of a random walk is known as the simple random walk, which is a stochastic process in discrete time with the integers as the state space, and is based on a Bernoulli process, where each Bernoulli variable takes either the value positive one or negative one.
An example of a model for such a field is the Ising model. A discrete-time stochastic process satisfying the ... But the state space would be of increasing ...
A continuous-time Markov chain (X t) t ≥ 0 is defined by a finite or countable state space S, a transition rate matrix Q with dimensions equal to that of the state space and initial probability distribution defined on the state space.
According to the figure, a bull week is followed by another bull week 90% of the time, a bear week 7.5% of the time, and a stagnant week the other 2.5% of the time. Labeling the state space {1 = bull, 2 = bear, 3 = stagnant} the transition matrix for this example is
A state i is inessential if it is not essential. [2] A state is final if and only if its communicating class is closed. A Markov chain is said to be irreducible if its state space is a single communicating class; in other words, if it is possible to get to any state from any state. [1] [3]: 20
Example of a simple MDP with three states (green circles) and two actions (orange circles), with two rewards (orange arrows) A Markov decision process is a 4-tuple (,,,), where: is a set of states called the state space. The state space may be discrete or continuous, like the set of real numbers.
In 1953 the term Markov chain was used for stochastic processes with discrete or continuous index set, living on a countable or finite state space, see Doob. [1] or Chung. [2] Since the late 20th century it became more popular to consider a Markov chain as a stochastic process with discrete index set, living on a measurable state space. [3] [4] [5]
A stochastic matrix describes a Markov chain X t over a finite state space S with cardinality α.. If the probability of moving from i to j in one time step is Pr(j|i) = P i,j, the stochastic matrix P is given by using P i,j as the i-th row and j-th column element, e.g.,