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all have the same [[probability distribution]]. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.<ref name="Lamperti1977page6">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=6 and 7}}</ref><ref name="GikhmanSkorokhod1969page4">{{cite book|author1=Iosif I. Gikhman |author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=4}}</ref> But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.<ref name="Lamperti1977page6"/><ref name="Adler2010page14">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA263|year=2010|publisher=SIAM|isbn=978-0-89871-693-1|pages=14, 15}}</ref><ref name="ChiuStoyan2013page112">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=112}}</ref>

all have the same [[probability distribution]]. The index set of a stationary stochastic process is usually interpreted as time, so it can be the integers or the real line.<ref name="Lamperti1977page6">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=6 and 7}}</ref><ref name="GikhmanSkorokhod1969page4">{{cite book|author1=Iosif I. Gikhman |author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=4}}</ref> But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.<ref name="Lamperti1977page6"/><ref name="Adler2010page14">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA263|year=2010|publisher=SIAM|isbn=978-0-89871-693-1|pages=14, 15}}</ref><ref name="ChiuStoyan2013page112">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=112}}</ref>

它们都有相同的[[概率分布]]。平稳随机过程的指标集通常被解释为时间，因此它可以是整数或实线=图书https://books.com/？id=pd4cvgaacaj | year=1977 | publisher=Springer Verlag | isbn=978-3-540-90275-1 | pages=6和7}</ref><ref name=“GikhmanSkorokhod1969page4”>{cite book | author1=Iosif I.Gikhman | author2=Anatoly Vladimirovich Skorokhod | title=随机过程理论简介|网址=图书https://books.com/？id=yJyLzG7N7r8C&pg=PR2 | year=1969 | publisher=Courier Corporation | isbn=978-0-486-69387-3 | page=4}}</ref>但是平稳性的概念也存在于点过程和随机域中，其中索引集不解释为时间。<ref name=“Lamperti1977page6”/><ref name=“Adler2010page14”>{cite book | author=Robert J.Adler | title=the Geometry of Random Fields | url=图书https://books.com/？id=ryejJmJAj28C&pg=PA263 | year=2010 2010 | publisher=SIAM | isbn=978-0-89871-693-1 |页数14，15}}</ref><ref name=“Chiustoyyan0101013Page112”{{{本书| author1=Sung Nok Chiu | author2=Dietrich Stoyan | author3=Wilfrid S.Kendall | author4=Joseph Meckee | title=随机几何及其应用其应用随机几何及其应用应用〈随机几何及其应用随机几何及其|网址=图书https://books.com/？id=825NfM6Nc EC |年份=2013 | publisher=John Wiley&Sons | isbn=978-1-118-65825-3 | page=112}</ref>

它们都有相同的[[概率分布]]。平稳随机过程的指标集通常被解释为时间，因此可以是整数或实线。<ref name="Lamperti1977page6">{{cite book|author=John Lamperti|title=Stochastic processes: a survey of the mathematical theory|url=https://books.google.com/books?id=Pd4cvgAACAAJ|year=1977|publisher=Springer-Verlag|isbn=978-3-540-90275-1|pages=6 and 7}}</ref><ref name="GikhmanSkorokhod1969page4">{{cite book|author1=Iosif I. Gikhman |author2=Anatoly Vladimirovich Skorokhod|title=Introduction to the Theory of Random Processes|url=https://books.google.com/books?id=yJyLzG7N7r8C&pg=PR2|year=1969|publisher=Courier Corporation|isbn=978-0-486-69387-3|page=4}}</ref> But the concept of stationarity also exists for point processes and random fields, where the index set is not interpreted as time.<ref name="Lamperti1977page6"/><ref name="Adler2010page14">{{cite book|author=Robert J. Adler|title=The Geometry of Random Fields|url=https://books.google.com/books?id=ryejJmJAj28C&pg=PA263|year=2010|publisher=SIAM|isbn=978-0-89871-693-1|pages=14, 15}}</ref><ref name="ChiuStoyan2013page112">{{cite book|author1=Sung Nok Chiu|author2=Dietrich Stoyan|author3=Wilfrid S. Kendall|author4=Joseph Mecke|title=Stochastic Geometry and Its Applications|url=https://books.google.com/books?id=825NfM6Nc-EC|year=2013|publisher=John Wiley & Sons|isbn=978-1-118-65825-3|page=112}}</ref>

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

A symmetric random walk and a Wiener process (with zero drift) are both examples of martingales, respectively, in discrete and continuous time. In this aspect, discrete-time martingales generalize the idea of partial sums of independent random variables.

在离散时间和连续时间中，对称随机游动和 Wiener 过程(带零漂)都是鞅的例子。在这方面，离散鞅推广了独立随机变量部分和的概念。

在离散时间和连续时间中，'''<font color="#ff8000"> 对称随机游动Symmetric random walk</font>'''和 维纳Wiener 过程(带零漂移)都是'''<font color="#ff8000"> 鞅Martingale</font>'''的例子。在这方面，离散'''<font color="#ff8000"> 鞅Martingale</font>'''推广了独立随机变量部分和的概念。

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.

Martingales can also be created from stochastic processes by applying some suitable transformations, which is the case for the homogeneous Poisson process (on the real line) resulting in a martingale called the compensated Poisson process.

也可以通过适当的变换从随机过程中产生鞅，这是齐次泊松过程(在实线上)产生一个被称为补偿泊松过程的鞅的情形。

也可以通过适当的变换从随机过程中产生'''<font color="#ff8000"> 鞅Martingale</font>'''，这是齐次泊松过程(在实线上)产生一个被称为补偿泊松过程的'''<font color="#ff8000"> 鞅Martingale</font>'''的情形。

Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.

Martingales mathematically formalize the idea of a fair game, and they were originally developed to show that it is not possible to win a fair game. Many problems in probability have been solved by finding a martingale in the problem and studying it. Martingales will converge, given some conditions on their moments, so they are often used to derive convergence results, due largely to martingale convergence theorems.

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