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第589行：
第589行： − + 第596行：
第596行： − “平稳性”是当随机过程的所有随机变量都是相同分布时随机过程所具有的数学性质。换言之，如果<math>X</math>是一个平稳随机过程，那么对于t</math>中的任何<math>t\In</math>随机变量，<math>n</math>具有相同的分布，这意味着对于任何一组<math>n</math>索引集值<math>t\u 1、\dots、t\n</math>而言，对应的<math>n</math>随机变量+ 第604行：
第604行： − 鞅是一个离散时间或连续时间的随机过程，其特性是，在给定过程的当前值和所有过去值的任何时刻，每个未来值的条件期望都等于当前值。在离散时间中，如果此属性对下一个值有效，则对所有未来值都有效。鞅的精确数学定义需要两个其他条件加上过滤的数学概念，这与随着时间的推移增加可用信息的直觉有关。鞅通常被定义为实值的，但是它们也可以是复值的，甚至是更一般的。+ 第636行：
第636行： − −
→Stationarity稳定性
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math>n</math>-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.
The concept of the Markov property was originally for stochastic processes in continuous and discrete time, but the property has been adapted for other index sets such as <math>n</math>-dimensional Euclidean space, which results in collections of random variables known as Markov random fields.
马尔可夫性的概念最初是用于连续和离散时间的随机过程，但是这个性质已经适用于其他指标集，如 < math > n </math > 维欧氏空间，这导致了被称为马尔可夫随机场的随机变量集合。
马尔可夫性的概念最初是用于连续和离散时间的随机过程，但是这个性质已经适用于其他指标集，如 <math>n</math>维欧氏空间，这导致了被称为马尔可夫随机场的随机变量集合。
{{Main|Stationary process}}
{{Main|Stationary process}}
'''Stationarity''' is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if <math>X</math> is a stationary stochastic process, then for any <math>t\in T</math> the random variable <math>X_t</math> has the same distribution, which means that for any set of <math>n</math> index set values <math>t_1,\dots, t_n</math>, the corresponding <math>n</math> random variables
'''Stationarity''' is a mathematical property that a stochastic process has when all the random variables of that stochastic process are identically distributed. In other words, if <math>X</math> is a stationary stochastic process, then for any <math>t\in T</math> the random variable <math>X_t</math> has the same distribution, which means that for any set of <math>n</math> index set values <math>t_1,\dots, t_n</math>, the corresponding <math>n</math> random variables
“稳定性”是当随机过程的所有随机变量都是相同分布时随机过程所具有的数学性质。换言之，如果<math>X</math>是一个平稳随机过程，那么对于任何<math>t\in T</math>，随机变量<math>X_t</math>具有相同的分布，这意味着对于任何一组<math>n</math>索引集值<math>t_1,\dots, t_n</math>而言，对应的<math>n</math>随机变量
<center><math>
<center><math>
A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.
A martingale is a discrete-time or continuous-time stochastic process with the property that, at every instant, given the current value and all the past values of the process, the conditional expectation of every future value is equal to the current value. In discrete time, if this property holds for the next value, then it holds for all future values. The exact mathematical definition of a martingale requires two other conditions coupled with the mathematical concept of a filtration, which is related to the intuition of increasing available information as time passes. Martingales are usually defined to be real-valued, but they can also be complex-valued or even more general.
'''<font color="#ff8000"> 鞅Martingale</font>'''是一个离散时间或连续时间的随机过程，其特性是，在给定过程的当前值和所有过去值的任何时刻，每个未来值的条件期望都等于当前值。在离散时间中，如果此属性对下一个值有效，则对所有未来值都有效。'''<font color="#ff8000"> 鞅Martingale</font>'''的精确数学定义需要两个其他条件加上过滤的数学概念，这与随着时间的推移增加可用信息的直觉有关。'''<font color="#ff8000"> 鞅Martingale</font>'''通常被定义为实值的，但是它们也可以取复值，甚至是更一般的值。
</math></center>
</math></center>
鞅数学形式化了公平游戏的概念，它们最初是为了证明不可能赢得公平游戏而开发的。通过在问题中找到一个鞅并研究它，已经解决了许多概率问题。由于鞅收敛定理的存在，在给定矩的一些条件下，鞅会收敛，因此常用它们来推导收敛结果。
鞅数学形式化了公平游戏的概念，它们最初是为了证明不可能赢得公平游戏而开发的。通过在问题中找到一个鞅并研究它，已经解决了许多概率问题。由于鞅收敛定理的存在，在给定矩的一些条件下，鞅会收敛，因此常用它们来推导收敛结果。
====Filtration过滤====
====Filtration过滤====