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where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <math>T</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page123">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=123}}</ref>

where <math>n\geq 1</math> is a counting number and each set <math>t_i</math> is a non-empty finite subset of the index set <math>T</math>, so each <math>t_i\subset T</math>, which means that <math>t_1,\dots,t_n</math> is any finite collection of subsets of the index set <math>T</math>.<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page123">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=123}}</ref>

其中<math>n\geq 1</math>是一个计数数，而每个集合<math>t\u i</math>是索引集<math>t</math>的一个非空有限子集，因此每个<math>t\i\子集t</math>，这意味着<math>t\u 1，tün</math>是索引集的任何子集的有限集合=图书https://books.com/？id=W0ydAgAAQBAJ&pg=PA356 | year=2000 | publisher=剑桥大学出版社| isbn=978-1-107-71749-7 | pages=123}</ref>

其中<math>n\geq 1</math>是一个计数数，而每个集合<math>t_i</math>>是索引集<math>T</math>的一个非空有限子集，因此每个<math>t_i\subset T</math>，这意味着<math>t_1,\dots,t_n</math>是索引集<math>T</math>的任何子集的有限集合<ref name="Kallenberg2002page24"/><ref name="RogersWilliams2000page123">{{cite book|author1=L. C. G. Rogers|author2=David Williams|title=Diffusions, Markov Processes, and Martingales: Volume 1, Foundations|url=https://books.google.com/books?id=W0ydAgAAQBAJ&pg=PA356|year=2000|publisher=Cambridge University Press|isbn=978-1-107-71749-7|pages=123}}</ref>id=W0ydAgAAQBAJ&pg=PA356 | year=2000 | publisher=剑桥大学出版社| isbn=978-1-107-71749-7 | pages=123}</ref>

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.

Markov processes are stochastic processes, traditionally in discrete or continuous time, that have the Markov property, which means the next value of the Markov process depends on the current value, but it is conditionally independent of the previous values of the stochastic process. In other words, the behavior of the process in the future is stochastically independent of its behavior in the past, given the current state of the process.

For any measurable subset <math>C</math> of the <math>n</math>-fold [[Cartesian power]] <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as:<ref name="Lamperti1977page1"/>

For any measurable subset <math>C</math> of the <math>n</math>-fold [[Cartesian power]] <math>S^n=S\times\dots \times S</math>, the finite-dimensional distributions of a stochastic process <math>X</math> can be written as:<ref name="Lamperti1977page1"/>

对于<math>n</math>-fold[[Cartesian power]]<math>S^n=S\times\dots\times</math>的任何可测子集，<math>X</math>的有限维分布可以写成：<ref name=“Lamperti1977page1”/>

 

对于<math>n</math>级[[笛卡尔幂]]<math>S^n=S\times\dots \times S</math>的任何可测子集<math>C</math>，<math>X</math>的有限维分布可以写成：<ref name=“Lamperti1977page1”/>

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.

The Brownian motion process and the Poisson process (in one dimension) are both examples of Markov processes in continuous time, while random walks on the integers and the gambler's ruin problem are examples of Markov processes in discrete time.

A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).

A Markov chain is a type of Markov process that has either discrete state space or discrete index set (often representing time), but the precise definition of a Markov chain varies. For example, it is common to define a Markov chain as a Markov process in either discrete or continuous time with a countable state space (thus regardless of the nature of time), but it has been also common to define a Markov chain as having discrete time in either countable or continuous state space (thus regardless of the state space).

马尔可夫链是一种具有离散状态空间或离散指标集(通常表示时间)的马尔可夫过程，但是马尔可夫链的精确定义是变化的。例如，通常将马尔可夫链定义为离散或连续时间中具有可数状态空间的马尔可夫过程(因此不考虑时间的性质) ，但也通常将马尔可夫链定义为在可数或连续状态空间中具有离散时间的马尔可夫链(因此不考虑状态空间)。

马尔可夫链是一种具有离散状态空间或离散指标集(通常表示时间)的马尔可夫过程，但是马尔可夫链的精确定义是变化的。例如，通常将'''<font color="#ff8000"> 马尔可夫链Markov chain</font>'''定义为离散或连续时间中具有可数状态空间的马尔可夫过程(因此不考虑时间的性质) ，但也通常将'''<font color="#ff8000"> 马尔可夫链Markov chain</font>'''定义为在可数或连续状态空间中具有离散时间的马尔可夫链(因此不考虑状态空间)。

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Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.

Markov processes form an important class of stochastic processes and have applications in many areas. For example, they are the basis for a general stochastic simulation method known as Markov chain Monte Carlo, which is used for simulating random objects with specific probability distributions, and has found application in Bayesian statistics.

 

 

====Stationarity稳定性====

====Stationarity稳定性====