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Two stochastic processes <math>X</math> and <math>Y</math> defined on the same probability space <math>(\Omega,\mathcal{F},P)</math> with the same index set <math>T</math> are said be '''independent''' if for all <math>n \in \mathbb{N}</math> and for every choice of epochs <math>t_1,\ldots,t_n \in T</math>, the random vectors <math>\left( X(t_1),\ldots,X(t_n) \right)</math> and <math>\left( Y(t_1),\ldots,Y(t_n) \right)</math> are independent.<ref name=Lapidoth>Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.</ref>{{rp|p. 515}}

Two stochastic processes <math>X</math> and <math>Y</math> defined on the same probability space <math>(\Omega,\mathcal{F},P)</math> with the same index set <math>T</math> are said be '''independent''' if for all <math>n \in \mathbb{N}</math> and for every choice of epochs <math>t_1,\ldots,t_n \in T</math>, the random vectors <math>\left( X(t_1),\ldots,X(t_n) \right)</math> and <math>\left( Y(t_1),\ldots,Y(t_n) \right)</math> are independent.<ref name=Lapidoth>Lapidoth, Amos, ''A Foundation in Digital Communication'', Cambridge University Press, 2009.</ref>{{rp|p. 515}}

两个在相同的概率空间<math>(\Omega,\mathcal{F},P)</math>上定义，具有相同索引集<math>T</math>的随机过程<math>X</math>和<math>Y</math>被称为“相互独立”，如果对于所有<math>n \in \mathbb{N}</math>，以及每个特定区间<math>t_1,\ldots,t_n \in T</math>中的随机向量<math>\left( X(t_1),\ldots,X(t_n) \right)</math> 和<math>\left( Y(t_1),\ldots,Y(t_n) \right)</math>是独立的。<ref name=Lapidoth>Amos，“数字通信基础”，剑桥大学出版社，2009年。</ref>{rp | p.515}}

两个在相同的概率空间<math>(\Omega,\mathcal{F},P)</math>上定义，具有相同索引集<math>T</math>的随机过程<math>X</math>和<math>Y</math>被称为“相互独立”，如果对于所有<math>n \in \mathbb{N}</math>，以及每个特定的<math>t_1,\ldots,t_n \in T</math>，随机向量<math>\left( X(t_1),\ldots,X(t_n) \right)</math> 和<math>\left( Y(t_1),\ldots,Y(t_n) \right)</math>是独立的。<ref name=Lapidoth>Amos，“数字通信基础”，剑桥大学出版社，2009年。</ref>{rp | p.515}}

In 1953 Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-André Meyer. Earlier work had been carried out by Sergei Bernstein, Paul Lévy and Jean Ville, the latter adopting the term martingale for the stochastic process. Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes. which would later result in Varadhan winning the 2007 Abel Prize. In the 1990s and 2000s the theories of Schramm–Loewner evolution and rough paths were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin Werner in 2008 and to Martin Hairer in 2014.

In 1953 Doob published his book Stochastic processes, which had a strong influence on the theory of stochastic processes and stressed the importance of measure theory in probability. Doob also chiefly developed the theory of martingales, with later substantial contributions by Paul-André Meyer. Earlier work had been carried out by Sergei Bernstein, Paul Lévy and Jean Ville, the latter adopting the term martingale for the stochastic process. Methods from the theory of martingales became popular for solving various probability problems. Techniques and theory were developed to study Markov processes and then applied to martingales. Conversely, methods from the theory of martingales were established to treat Markov processes. which would later result in Varadhan winning the 2007 Abel Prize. In the 1990s and 2000s the theories of Schramm–Loewner evolution and rough paths were introduced and developed to study stochastic processes and other mathematical objects in probability theory, which respectively resulted in Fields Medals being awarded to Wendelin Werner in 2008 and to Martin Hairer in 2014.