Skip to main content
Cross Validated

Return to Answer

added 35 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ is measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $\{(X_n, \mathscr{F}_n)\}$ is a martingale, then $\{(X_n, \mathscr{G}_n)\}$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ areis measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$) larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That saidIn other words, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, -- it may cover any information up to time $n$ as long as Condition 4 holds.
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ is measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $\{(X_n, \mathscr{F}_n)\}$ is a martingale, then $\{(X_n, \mathscr{G}_n)\}$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ is measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $\{(X_n, \mathscr{F}_n)\}$ is a martingale, then $\{(X_n, \mathscr{G}_n)\}$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ is measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". In other words, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself -- it may cover any information up to time $n$ as long as Condition 4 holds.
added 7 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ areis measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $(X_n, \mathscr{F}_n)$$\{(X_n, \mathscr{F}_n)\}$ is a martingale, then $(X_n, \mathscr{G}_n)$$\{(X_n, \mathscr{G}_n)\}$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $(X_n, \mathscr{F}_n)$ is a martingale, then $(X_n, \mathscr{G}_n)$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ is measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $\{(X_n, \mathscr{F}_n)\}$ is a martingale, then $\{(X_n, \mathscr{G}_n)\}$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
deleted 3 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, they areit is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $(X_n, \mathscr{F}_n)$ is a martingale, then $(X_n, \mathscr{G}_n)$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, they are just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $(X_n, \mathscr{F}_n)$ is a martingale, then $(X_n, \mathscr{G}_n)$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
  1. The "$X_1, \ldots, X_n$" in the notation "$E[X_{n + 1}|X_1, \ldots, X_n]$" should be interpreted as the $\sigma$-field $\sigma(X_1, \ldots, X_n)$, instead of $n$ isolated random variables. In general, "$E[X|Y]$" is a shorthand for the measure-theoretic conditional expectation $E[X|\sigma(Y)]$. The $\sigma$-field $\sigma(X_1, \ldots, X_n)$, known as the $\sigma$-field generated by the random vector $(X_1, \ldots, X_n)$, is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable. Therefore, while your statement "the set of past history $\{X_1, \ldots, X_n\}$ is not a sigma-field" is trivially true (written in this way, it is just a collection of $n$ random variables), it should not be interpreted in this way when they appeared in equation $(2)$.
  2. As the quotation block containing equation $(2)$ demonstrates, the Wikipedia's definition is indeed "nested" in Durrett's definition: $\mathscr{G}_n := \sigma(X_1, \ldots, X_n)$ is just one special filtration satisfying Condition 1 and Condition 2. Furthermore, $\mathscr{G}_n$ are the smallest $\sigma$-fields with respect to which the $X_n$ are a martingale. That is, suppose that there exists a filtration $\{\mathscr{F}_n\}$ such that $(X_n, \mathscr{F}_n)$ is a martingale, then $(X_n, \mathscr{G}_n)$ must be a martingale as well and $\mathscr{G}_n \subset \mathscr{F}_n$ for each $n$ (recall that in the last bullet, I mentioned that $\mathscr{G}_n$ is the smallest $\sigma$-field in $\mathscr{F}$ with respect to which $(X_1, \ldots, X_n)$ are measurable). For this reason, the filtration $\{\mathscr{G}_n\}$ is referred as a natural filtration in some literature.
  3. At this point, it should be clear to you that the role of $\sigma$-fields $\mathscr{F}_n$ in martingale's definition is essential, for with the same sequence of random variables $\{X_n\}$, different martingales can be constructed by choosing different filtrations with respect to which $X_n$ are measurable. See Example 35.1 in Billingsley's book for a concrete example, in which he wrote "It is natural and convenient to allow the $\sigma$-fields $\mathscr{F}_n$ larger than the minimal ones ($\sigma(X_1, \ldots, X_n)$)". That said, the "past history" may well be richer than the sequence of $\{X_1, \ldots, X_n\}$ itself, it may cover any information up to time $n$.
added 70 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
Reorganize the writing.
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
edited body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
added 195 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
added 243 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
added 3 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
added 140 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
added 400 characters in body
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89
Source Link
Zhanxiong
Zhanxiong
  • 21.2k
  • 2
  • 46
  • 89