Proving a sequence forms a martingale












1












$begingroup$


Let $Omega = mathbb N = {1,2,3,cdots}$ and $mathscr F_n$ be the $sigma$-field generated by the sets ${1},{2},cdots,{[n+1,infty)}$



Define a probability on $mathbb N$ by setting $mathbb P([n,infty)) = frac 1 n$



Show that



1) $f_n =(n+1)mathbf1_{[n+1,infty)}$ is a martingale

2) $f_n to 0 quad a.s.$



I'm having difficulty dealing with the indicator r.v. How do we get it out of the conditional expectation for $mathbf E[f_{n+1}mid mathcal{F}_n]$?



I tried to graph it out and I think I can see why it forms a martingale but I'm not able to prove it explicitly.



<span class=$f_n$">






probability-theory martingales







share|cite|improve this question











$endgroup$












  • $begingroup$
    @saz, Any luck with this question?
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 15:08










  • $begingroup$
    I know how to solve this question, yes. But since you don't show any effort, I don't feel motivated to answer it.
    $endgroup$
    – saz
    Apr 27 '15 at 16:18










  • $begingroup$
    @saz, I have tried two ways to show it but I don't know if they're the right paths. The first attempt was to take $mathbb E[f_{n+1} lvert mathscr F_n]$ and split $[n+1,infty)$ into ${n} cup [n+1,infty)$ but the indicator variable is still within the expectation. My next attempt was to graph it out but I still can't get to the answer. I am truly stuck at this point. A pointer in the right direction would help.
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 17:16










  • $begingroup$
    So, any progress?
    $endgroup$
    – saz
    Apr 28 '15 at 18:10










  • $begingroup$
    @saz I sort of follow what you're saying but we never used $mathscr G$ in our study. My course is "classical probability" and tends to avoid measure theory. I'm still working out step 1. I'll update once I wrap my head around it.
    $endgroup$
    – Srinivas Eswar
    Apr 28 '15 at 22:19


















1












$begingroup$


Let $Omega = mathbb N = {1,2,3,cdots}$ and $mathscr F_n$ be the $sigma$-field generated by the sets ${1},{2},cdots,{[n+1,infty)}$



Define a probability on $mathbb N$ by setting $mathbb P([n,infty)) = frac 1 n$



Show that



1) $f_n =(n+1)mathbf1_{[n+1,infty)}$ is a martingale

2) $f_n to 0 quad a.s.$



I'm having difficulty dealing with the indicator r.v. How do we get it out of the conditional expectation for $mathbf E[f_{n+1}mid mathcal{F}_n]$?



I tried to graph it out and I think I can see why it forms a martingale but I'm not able to prove it explicitly.



<span class=$f_n$">






probability-theory martingales







share|cite|improve this question











$endgroup$












  • $begingroup$
    @saz, Any luck with this question?
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 15:08










  • $begingroup$
    I know how to solve this question, yes. But since you don't show any effort, I don't feel motivated to answer it.
    $endgroup$
    – saz
    Apr 27 '15 at 16:18










  • $begingroup$
    @saz, I have tried two ways to show it but I don't know if they're the right paths. The first attempt was to take $mathbb E[f_{n+1} lvert mathscr F_n]$ and split $[n+1,infty)$ into ${n} cup [n+1,infty)$ but the indicator variable is still within the expectation. My next attempt was to graph it out but I still can't get to the answer. I am truly stuck at this point. A pointer in the right direction would help.
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 17:16










  • $begingroup$
    So, any progress?
    $endgroup$
    – saz
    Apr 28 '15 at 18:10










  • $begingroup$
    @saz I sort of follow what you're saying but we never used $mathscr G$ in our study. My course is "classical probability" and tends to avoid measure theory. I'm still working out step 1. I'll update once I wrap my head around it.
    $endgroup$
    – Srinivas Eswar
    Apr 28 '15 at 22:19
















1












1








1





$begingroup$


Let $Omega = mathbb N = {1,2,3,cdots}$ and $mathscr F_n$ be the $sigma$-field generated by the sets ${1},{2},cdots,{[n+1,infty)}$



Define a probability on $mathbb N$ by setting $mathbb P([n,infty)) = frac 1 n$



Show that



1) $f_n =(n+1)mathbf1_{[n+1,infty)}$ is a martingale

2) $f_n to 0 quad a.s.$



I'm having difficulty dealing with the indicator r.v. How do we get it out of the conditional expectation for $mathbf E[f_{n+1}mid mathcal{F}_n]$?



I tried to graph it out and I think I can see why it forms a martingale but I'm not able to prove it explicitly.



<span class=$f_n$">






probability-theory martingales







share|cite|improve this question











$endgroup$




Let $Omega = mathbb N = {1,2,3,cdots}$ and $mathscr F_n$ be the $sigma$-field generated by the sets ${1},{2},cdots,{[n+1,infty)}$



Define a probability on $mathbb N$ by setting $mathbb P([n,infty)) = frac 1 n$



Show that



1) $f_n =(n+1)mathbf1_{[n+1,infty)}$ is a martingale

2) $f_n to 0 quad a.s.$



I'm having difficulty dealing with the indicator r.v. How do we get it out of the conditional expectation for $mathbf E[f_{n+1}mid mathcal{F}_n]$?



I tried to graph it out and I think I can see why it forms a martingale but I'm not able to prove it explicitly.



<span class=$f_n$">






probability-theory martingales




probability-theory martingales






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 5 at 20:06









saz

81.5k861127




81.5k861127










asked Apr 22 '15 at 21:44









Srinivas EswarSrinivas Eswar

11911




11911












  • $begingroup$
    @saz, Any luck with this question?
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 15:08










  • $begingroup$
    I know how to solve this question, yes. But since you don't show any effort, I don't feel motivated to answer it.
    $endgroup$
    – saz
    Apr 27 '15 at 16:18










  • $begingroup$
    @saz, I have tried two ways to show it but I don't know if they're the right paths. The first attempt was to take $mathbb E[f_{n+1} lvert mathscr F_n]$ and split $[n+1,infty)$ into ${n} cup [n+1,infty)$ but the indicator variable is still within the expectation. My next attempt was to graph it out but I still can't get to the answer. I am truly stuck at this point. A pointer in the right direction would help.
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 17:16










  • $begingroup$
    So, any progress?
    $endgroup$
    – saz
    Apr 28 '15 at 18:10










  • $begingroup$
    @saz I sort of follow what you're saying but we never used $mathscr G$ in our study. My course is "classical probability" and tends to avoid measure theory. I'm still working out step 1. I'll update once I wrap my head around it.
    $endgroup$
    – Srinivas Eswar
    Apr 28 '15 at 22:19




















  • $begingroup$
    @saz, Any luck with this question?
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 15:08










  • $begingroup$
    I know how to solve this question, yes. But since you don't show any effort, I don't feel motivated to answer it.
    $endgroup$
    – saz
    Apr 27 '15 at 16:18










  • $begingroup$
    @saz, I have tried two ways to show it but I don't know if they're the right paths. The first attempt was to take $mathbb E[f_{n+1} lvert mathscr F_n]$ and split $[n+1,infty)$ into ${n} cup [n+1,infty)$ but the indicator variable is still within the expectation. My next attempt was to graph it out but I still can't get to the answer. I am truly stuck at this point. A pointer in the right direction would help.
    $endgroup$
    – Srinivas Eswar
    Apr 27 '15 at 17:16










  • $begingroup$
    So, any progress?
    $endgroup$
    – saz
    Apr 28 '15 at 18:10










  • $begingroup$
    @saz I sort of follow what you're saying but we never used $mathscr G$ in our study. My course is "classical probability" and tends to avoid measure theory. I'm still working out step 1. I'll update once I wrap my head around it.
    $endgroup$
    – Srinivas Eswar
    Apr 28 '15 at 22:19


















$begingroup$
@saz, Any luck with this question?
$endgroup$
– Srinivas Eswar
Apr 27 '15 at 15:08




$begingroup$
@saz, Any luck with this question?
$endgroup$
– Srinivas Eswar
Apr 27 '15 at 15:08












$begingroup$
I know how to solve this question, yes. But since you don't show any effort, I don't feel motivated to answer it.
$endgroup$
– saz
Apr 27 '15 at 16:18




$begingroup$
I know how to solve this question, yes. But since you don't show any effort, I don't feel motivated to answer it.
$endgroup$
– saz
Apr 27 '15 at 16:18












$begingroup$
@saz, I have tried two ways to show it but I don't know if they're the right paths. The first attempt was to take $mathbb E[f_{n+1} lvert mathscr F_n]$ and split $[n+1,infty)$ into ${n} cup [n+1,infty)$ but the indicator variable is still within the expectation. My next attempt was to graph it out but I still can't get to the answer. I am truly stuck at this point. A pointer in the right direction would help.
$endgroup$
– Srinivas Eswar
Apr 27 '15 at 17:16




$begingroup$
@saz, I have tried two ways to show it but I don't know if they're the right paths. The first attempt was to take $mathbb E[f_{n+1} lvert mathscr F_n]$ and split $[n+1,infty)$ into ${n} cup [n+1,infty)$ but the indicator variable is still within the expectation. My next attempt was to graph it out but I still can't get to the answer. I am truly stuck at this point. A pointer in the right direction would help.
$endgroup$
– Srinivas Eswar
Apr 27 '15 at 17:16












$begingroup$
So, any progress?
$endgroup$
– saz
Apr 28 '15 at 18:10




$begingroup$
So, any progress?
$endgroup$
– saz
Apr 28 '15 at 18:10












$begingroup$
@saz I sort of follow what you're saying but we never used $mathscr G$ in our study. My course is "classical probability" and tends to avoid measure theory. I'm still working out step 1. I'll update once I wrap my head around it.
$endgroup$
– Srinivas Eswar
Apr 28 '15 at 22:19






$begingroup$
@saz I sort of follow what you're saying but we never used $mathscr G$ in our study. My course is "classical probability" and tends to avoid measure theory. I'm still working out step 1. I'll update once I wrap my head around it.
$endgroup$
– Srinivas Eswar
Apr 28 '15 at 22:19












1 Answer
1






active

oldest

votes


















2












$begingroup$

Hints:




  1. It is well-known that a random variable $Y in L^1$ equals $mathbb{E}(X mid mathcal{F})$ if and only if it is $mathcal{F}$-measurable and $$int_G Y , dmathbb{P} = int_G X , dmathbb{P} qquad text{for all $G in mathcal{G}$}$$ where $mathcal{G}$ denotes a $cap$-stable generator of $mathcal{F}$, i.e. $sigma(mathcal{G}) = mathcal{F}$ and $$G_1, G_2 in mathcal{G} Rightarrow G_1 cap G_2 in mathcal{G}.$$ Check that $mathcal{G}_n := {emptyset,{1},{2},ldots,[n+1,infty)}$ is a $cap$-stable generator of $mathcal{F}_n$.

  2. In order to show that $f_n$ is a martingale, it suffices to show $$mathbb{E}(f_n mid mathcal{F}_{n-1}) = f_{n-1};$$ by step 1 this is equivalent to $$int_G f_n , dmathbb{P} = int_G f_{n-1} , dmathbb{P} tag{1}$$ for all $G in mathcal{G}_{n-1}$.

  3. By the very definition of $f_n$, we have $$begin{align*} A&:= int_G f_n , dmathbb{P} = (n+1) mathbb{P}(G cap [n+1,infty)) \ B &:= int_G f_{n-1} , dmathbb{P} = n mathbb{P}(G cap [n,infty)). end{align*}$$ We have to prove $A=B$ for all $G in mathcal{G}_{n-1}$. Consider two cases separately:

    • $G = {j}$ for some $j in {1,ldots,n-1}$: Since $G cap [ninfty) = G cap [n+1,infty)=emptyset$, we get $A=B=0$.

    • $G = [n,infty)$: Conclude from $G cap [n,infty)= [n,infty)$ and $G cap [n+1,infty) = [n+1,infty)$ that $A=B=1$.








share|cite|improve this answer











$endgroup$













  • $begingroup$
    I finally got back to this guy. I'm still not clear on a lot of things here. It's basically step 1. If I accept that I'm able to follow steps 2-3. Can you point me to some sources where I can read up on step 1. I can see how the addition of $phi$ has made $mathscr{G_n}$ $cap$-stable but I don't know anything about $L^1$ spaces or why this result (i.e. step 1) holds.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 18:36












  • $begingroup$
    @SrinivasEswar What $phi$ are you talking about? Do you know the result for $mathcal{G} := mathcal{F}$? In this case, it is a standard characterization of cond. expectations which you can find in any textbook on this topic.
    $endgroup$
    – saz
    Aug 28 '15 at 22:15












  • $begingroup$
    I was referring to the null set. Thanks for the help.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 23:45











Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1247342%2fproving-a-sequence-forms-a-martingale%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

Hints:




  1. It is well-known that a random variable $Y in L^1$ equals $mathbb{E}(X mid mathcal{F})$ if and only if it is $mathcal{F}$-measurable and $$int_G Y , dmathbb{P} = int_G X , dmathbb{P} qquad text{for all $G in mathcal{G}$}$$ where $mathcal{G}$ denotes a $cap$-stable generator of $mathcal{F}$, i.e. $sigma(mathcal{G}) = mathcal{F}$ and $$G_1, G_2 in mathcal{G} Rightarrow G_1 cap G_2 in mathcal{G}.$$ Check that $mathcal{G}_n := {emptyset,{1},{2},ldots,[n+1,infty)}$ is a $cap$-stable generator of $mathcal{F}_n$.

  2. In order to show that $f_n$ is a martingale, it suffices to show $$mathbb{E}(f_n mid mathcal{F}_{n-1}) = f_{n-1};$$ by step 1 this is equivalent to $$int_G f_n , dmathbb{P} = int_G f_{n-1} , dmathbb{P} tag{1}$$ for all $G in mathcal{G}_{n-1}$.

  3. By the very definition of $f_n$, we have $$begin{align*} A&:= int_G f_n , dmathbb{P} = (n+1) mathbb{P}(G cap [n+1,infty)) \ B &:= int_G f_{n-1} , dmathbb{P} = n mathbb{P}(G cap [n,infty)). end{align*}$$ We have to prove $A=B$ for all $G in mathcal{G}_{n-1}$. Consider two cases separately:

    • $G = {j}$ for some $j in {1,ldots,n-1}$: Since $G cap [ninfty) = G cap [n+1,infty)=emptyset$, we get $A=B=0$.

    • $G = [n,infty)$: Conclude from $G cap [n,infty)= [n,infty)$ and $G cap [n+1,infty) = [n+1,infty)$ that $A=B=1$.








share|cite|improve this answer











$endgroup$













  • $begingroup$
    I finally got back to this guy. I'm still not clear on a lot of things here. It's basically step 1. If I accept that I'm able to follow steps 2-3. Can you point me to some sources where I can read up on step 1. I can see how the addition of $phi$ has made $mathscr{G_n}$ $cap$-stable but I don't know anything about $L^1$ spaces or why this result (i.e. step 1) holds.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 18:36












  • $begingroup$
    @SrinivasEswar What $phi$ are you talking about? Do you know the result for $mathcal{G} := mathcal{F}$? In this case, it is a standard characterization of cond. expectations which you can find in any textbook on this topic.
    $endgroup$
    – saz
    Aug 28 '15 at 22:15












  • $begingroup$
    I was referring to the null set. Thanks for the help.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 23:45
















2












$begingroup$

Hints:




  1. It is well-known that a random variable $Y in L^1$ equals $mathbb{E}(X mid mathcal{F})$ if and only if it is $mathcal{F}$-measurable and $$int_G Y , dmathbb{P} = int_G X , dmathbb{P} qquad text{for all $G in mathcal{G}$}$$ where $mathcal{G}$ denotes a $cap$-stable generator of $mathcal{F}$, i.e. $sigma(mathcal{G}) = mathcal{F}$ and $$G_1, G_2 in mathcal{G} Rightarrow G_1 cap G_2 in mathcal{G}.$$ Check that $mathcal{G}_n := {emptyset,{1},{2},ldots,[n+1,infty)}$ is a $cap$-stable generator of $mathcal{F}_n$.

  2. In order to show that $f_n$ is a martingale, it suffices to show $$mathbb{E}(f_n mid mathcal{F}_{n-1}) = f_{n-1};$$ by step 1 this is equivalent to $$int_G f_n , dmathbb{P} = int_G f_{n-1} , dmathbb{P} tag{1}$$ for all $G in mathcal{G}_{n-1}$.

  3. By the very definition of $f_n$, we have $$begin{align*} A&:= int_G f_n , dmathbb{P} = (n+1) mathbb{P}(G cap [n+1,infty)) \ B &:= int_G f_{n-1} , dmathbb{P} = n mathbb{P}(G cap [n,infty)). end{align*}$$ We have to prove $A=B$ for all $G in mathcal{G}_{n-1}$. Consider two cases separately:

    • $G = {j}$ for some $j in {1,ldots,n-1}$: Since $G cap [ninfty) = G cap [n+1,infty)=emptyset$, we get $A=B=0$.

    • $G = [n,infty)$: Conclude from $G cap [n,infty)= [n,infty)$ and $G cap [n+1,infty) = [n+1,infty)$ that $A=B=1$.








share|cite|improve this answer











$endgroup$













  • $begingroup$
    I finally got back to this guy. I'm still not clear on a lot of things here. It's basically step 1. If I accept that I'm able to follow steps 2-3. Can you point me to some sources where I can read up on step 1. I can see how the addition of $phi$ has made $mathscr{G_n}$ $cap$-stable but I don't know anything about $L^1$ spaces or why this result (i.e. step 1) holds.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 18:36












  • $begingroup$
    @SrinivasEswar What $phi$ are you talking about? Do you know the result for $mathcal{G} := mathcal{F}$? In this case, it is a standard characterization of cond. expectations which you can find in any textbook on this topic.
    $endgroup$
    – saz
    Aug 28 '15 at 22:15












  • $begingroup$
    I was referring to the null set. Thanks for the help.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 23:45














2












2








2





$begingroup$

Hints:




  1. It is well-known that a random variable $Y in L^1$ equals $mathbb{E}(X mid mathcal{F})$ if and only if it is $mathcal{F}$-measurable and $$int_G Y , dmathbb{P} = int_G X , dmathbb{P} qquad text{for all $G in mathcal{G}$}$$ where $mathcal{G}$ denotes a $cap$-stable generator of $mathcal{F}$, i.e. $sigma(mathcal{G}) = mathcal{F}$ and $$G_1, G_2 in mathcal{G} Rightarrow G_1 cap G_2 in mathcal{G}.$$ Check that $mathcal{G}_n := {emptyset,{1},{2},ldots,[n+1,infty)}$ is a $cap$-stable generator of $mathcal{F}_n$.

  2. In order to show that $f_n$ is a martingale, it suffices to show $$mathbb{E}(f_n mid mathcal{F}_{n-1}) = f_{n-1};$$ by step 1 this is equivalent to $$int_G f_n , dmathbb{P} = int_G f_{n-1} , dmathbb{P} tag{1}$$ for all $G in mathcal{G}_{n-1}$.

  3. By the very definition of $f_n$, we have $$begin{align*} A&:= int_G f_n , dmathbb{P} = (n+1) mathbb{P}(G cap [n+1,infty)) \ B &:= int_G f_{n-1} , dmathbb{P} = n mathbb{P}(G cap [n,infty)). end{align*}$$ We have to prove $A=B$ for all $G in mathcal{G}_{n-1}$. Consider two cases separately:

    • $G = {j}$ for some $j in {1,ldots,n-1}$: Since $G cap [ninfty) = G cap [n+1,infty)=emptyset$, we get $A=B=0$.

    • $G = [n,infty)$: Conclude from $G cap [n,infty)= [n,infty)$ and $G cap [n+1,infty) = [n+1,infty)$ that $A=B=1$.








share|cite|improve this answer











$endgroup$



Hints:




  1. It is well-known that a random variable $Y in L^1$ equals $mathbb{E}(X mid mathcal{F})$ if and only if it is $mathcal{F}$-measurable and $$int_G Y , dmathbb{P} = int_G X , dmathbb{P} qquad text{for all $G in mathcal{G}$}$$ where $mathcal{G}$ denotes a $cap$-stable generator of $mathcal{F}$, i.e. $sigma(mathcal{G}) = mathcal{F}$ and $$G_1, G_2 in mathcal{G} Rightarrow G_1 cap G_2 in mathcal{G}.$$ Check that $mathcal{G}_n := {emptyset,{1},{2},ldots,[n+1,infty)}$ is a $cap$-stable generator of $mathcal{F}_n$.

  2. In order to show that $f_n$ is a martingale, it suffices to show $$mathbb{E}(f_n mid mathcal{F}_{n-1}) = f_{n-1};$$ by step 1 this is equivalent to $$int_G f_n , dmathbb{P} = int_G f_{n-1} , dmathbb{P} tag{1}$$ for all $G in mathcal{G}_{n-1}$.

  3. By the very definition of $f_n$, we have $$begin{align*} A&:= int_G f_n , dmathbb{P} = (n+1) mathbb{P}(G cap [n+1,infty)) \ B &:= int_G f_{n-1} , dmathbb{P} = n mathbb{P}(G cap [n,infty)). end{align*}$$ We have to prove $A=B$ for all $G in mathcal{G}_{n-1}$. Consider two cases separately:

    • $G = {j}$ for some $j in {1,ldots,n-1}$: Since $G cap [ninfty) = G cap [n+1,infty)=emptyset$, we get $A=B=0$.

    • $G = [n,infty)$: Conclude from $G cap [n,infty)= [n,infty)$ and $G cap [n+1,infty) = [n+1,infty)$ that $A=B=1$.









share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Apr 28 '15 at 17:30

























answered Apr 27 '15 at 18:24









sazsaz

81.5k861127




81.5k861127












  • $begingroup$
    I finally got back to this guy. I'm still not clear on a lot of things here. It's basically step 1. If I accept that I'm able to follow steps 2-3. Can you point me to some sources where I can read up on step 1. I can see how the addition of $phi$ has made $mathscr{G_n}$ $cap$-stable but I don't know anything about $L^1$ spaces or why this result (i.e. step 1) holds.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 18:36












  • $begingroup$
    @SrinivasEswar What $phi$ are you talking about? Do you know the result for $mathcal{G} := mathcal{F}$? In this case, it is a standard characterization of cond. expectations which you can find in any textbook on this topic.
    $endgroup$
    – saz
    Aug 28 '15 at 22:15












  • $begingroup$
    I was referring to the null set. Thanks for the help.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 23:45


















  • $begingroup$
    I finally got back to this guy. I'm still not clear on a lot of things here. It's basically step 1. If I accept that I'm able to follow steps 2-3. Can you point me to some sources where I can read up on step 1. I can see how the addition of $phi$ has made $mathscr{G_n}$ $cap$-stable but I don't know anything about $L^1$ spaces or why this result (i.e. step 1) holds.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 18:36












  • $begingroup$
    @SrinivasEswar What $phi$ are you talking about? Do you know the result for $mathcal{G} := mathcal{F}$? In this case, it is a standard characterization of cond. expectations which you can find in any textbook on this topic.
    $endgroup$
    – saz
    Aug 28 '15 at 22:15












  • $begingroup$
    I was referring to the null set. Thanks for the help.
    $endgroup$
    – Srinivas Eswar
    Aug 28 '15 at 23:45
















$begingroup$
I finally got back to this guy. I'm still not clear on a lot of things here. It's basically step 1. If I accept that I'm able to follow steps 2-3. Can you point me to some sources where I can read up on step 1. I can see how the addition of $phi$ has made $mathscr{G_n}$ $cap$-stable but I don't know anything about $L^1$ spaces or why this result (i.e. step 1) holds.
$endgroup$
– Srinivas Eswar
Aug 28 '15 at 18:36






$begingroup$
I finally got back to this guy. I'm still not clear on a lot of things here. It's basically step 1. If I accept that I'm able to follow steps 2-3. Can you point me to some sources where I can read up on step 1. I can see how the addition of $phi$ has made $mathscr{G_n}$ $cap$-stable but I don't know anything about $L^1$ spaces or why this result (i.e. step 1) holds.
$endgroup$
– Srinivas Eswar
Aug 28 '15 at 18:36














$begingroup$
@SrinivasEswar What $phi$ are you talking about? Do you know the result for $mathcal{G} := mathcal{F}$? In this case, it is a standard characterization of cond. expectations which you can find in any textbook on this topic.
$endgroup$
– saz
Aug 28 '15 at 22:15






$begingroup$
@SrinivasEswar What $phi$ are you talking about? Do you know the result for $mathcal{G} := mathcal{F}$? In this case, it is a standard characterization of cond. expectations which you can find in any textbook on this topic.
$endgroup$
– saz
Aug 28 '15 at 22:15














$begingroup$
I was referring to the null set. Thanks for the help.
$endgroup$
– Srinivas Eswar
Aug 28 '15 at 23:45




$begingroup$
I was referring to the null set. Thanks for the help.
$endgroup$
– Srinivas Eswar
Aug 28 '15 at 23:45


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


To learn more, see our tips on writing great answers.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f1247342%2fproving-a-sequence-forms-a-martingale%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Ellipse (mathématiques)

Image
Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
Read more

Quarter-circle Tiles

Image
up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
Read more

Mont Emei

Image
Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
Read more