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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 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
While the time parameter is usually discrete, the state space of a Markov chain does not have any generally agreed-on restrictions: the term may refer to a process on an arbitrary state space. [15] However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis.
If the state space is the real line, then the stochastic process is referred to as a real-valued stochastic process or a process with continuous state space. If the state space is n {\displaystyle n} -dimensional Euclidean space, then the stochastic process is called a n {\displaystyle n} - dimensional vector process or n {\displaystyle n ...
Let (Ω, Σ, P) be a probability space, let T be some interval of time, and let X : T × Ω → S be a stochastic process. For simplicity, the rest of this article will take the state space S to be the real line R, but the definitions go through mutatis mutandis if S is R n, a normed vector space, or even a general metric space.
To see the difference, consider the probability for a certain event in the game. In the above-mentioned dice games, the only thing that matters is the current state of the board. The next state of the board depends on the current state, and the next roll of the dice. It does not depend on how things got to their current state.
In mathematics, progressive measurability is a property in the theory of stochastic processes. A progressively measurable process, while defined quite technically, is important because it implies the stopped process is measurable. Being progressively measurable is a strictly stronger property than the notion of being an adapted process. [1]
A stochastic process has the Markov property if the conditional probability distribution of future states of the process (conditional on both past and present values) depends only upon the present state; that is, given the present, the future does not depend on the past.