Equivalence of the condition that the supremum of i.i.d. RVs are finite a.s.












1












$begingroup$


I am proving the following :



Suppose ${X_n : ninmathbb{N}}$ are i.i.d. random variables. Then $P(sup_{ninmathbb{N}}X_n < infty) = 1$ if and only if $ sum_{ninmathbb{N}}{P(X_n > M)} < infty$ for some $M>0.$



I guess by the form of the problem, that Borel-Canteli lemma will be used at some point, but I could not thought of a good way to connect it to the problem.



Any idea or hint will be really helpful. Thanks.



(edit)



Assume $sum_{ninmathbb{N}}{P(X_n > M)} < infty.$ In fact, as they are i.i.d., $P(X_n > M) = P(X_1>M)$ for all $n.$ Thus $P(X_1>M) = 0$ and $P(sup_{ninmathbb{N}}X_n > M) le sum_{ninmathbb{N}}{P(X_n > M)} = 0.$



Conversely, I've realized that ${sup_{ninmathbb{N}}{X_n} < infty} = bigcup_{m=1}^{infty}bigcap_{n=1}^{infty}{X_n le m}.$






probability-theory measure-theory random-variables independence borel-cantelli-lemmas







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Are you sure the random variables $(X_n)$ are supposed to be i.i.d., or only independent?
    $endgroup$
    – Did
    Dec 10 '18 at 15:20










  • $begingroup$
    I am 100 % sure that they are supposed to be i.i.d., as it was exam problem.
    $endgroup$
    – Euduardo
    Dec 10 '18 at 15:22










  • $begingroup$
    Well then the exam is slightly absurd, but why not.
    $endgroup$
    – Did
    Dec 10 '18 at 15:23










  • $begingroup$
    You're right Did. Now I understand independence is enough, thanks!
    $endgroup$
    – Euduardo
    Dec 11 '18 at 5:09
















1












$begingroup$


I am proving the following :



Suppose ${X_n : ninmathbb{N}}$ are i.i.d. random variables. Then $P(sup_{ninmathbb{N}}X_n < infty) = 1$ if and only if $ sum_{ninmathbb{N}}{P(X_n > M)} < infty$ for some $M>0.$



I guess by the form of the problem, that Borel-Canteli lemma will be used at some point, but I could not thought of a good way to connect it to the problem.



Any idea or hint will be really helpful. Thanks.



(edit)



Assume $sum_{ninmathbb{N}}{P(X_n > M)} < infty.$ In fact, as they are i.i.d., $P(X_n > M) = P(X_1>M)$ for all $n.$ Thus $P(X_1>M) = 0$ and $P(sup_{ninmathbb{N}}X_n > M) le sum_{ninmathbb{N}}{P(X_n > M)} = 0.$



Conversely, I've realized that ${sup_{ninmathbb{N}}{X_n} < infty} = bigcup_{m=1}^{infty}bigcap_{n=1}^{infty}{X_n le m}.$






probability-theory measure-theory random-variables independence borel-cantelli-lemmas







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Are you sure the random variables $(X_n)$ are supposed to be i.i.d., or only independent?
    $endgroup$
    – Did
    Dec 10 '18 at 15:20










  • $begingroup$
    I am 100 % sure that they are supposed to be i.i.d., as it was exam problem.
    $endgroup$
    – Euduardo
    Dec 10 '18 at 15:22










  • $begingroup$
    Well then the exam is slightly absurd, but why not.
    $endgroup$
    – Did
    Dec 10 '18 at 15:23










  • $begingroup$
    You're right Did. Now I understand independence is enough, thanks!
    $endgroup$
    – Euduardo
    Dec 11 '18 at 5:09














1












1








1





$begingroup$


I am proving the following :



Suppose ${X_n : ninmathbb{N}}$ are i.i.d. random variables. Then $P(sup_{ninmathbb{N}}X_n < infty) = 1$ if and only if $ sum_{ninmathbb{N}}{P(X_n > M)} < infty$ for some $M>0.$



I guess by the form of the problem, that Borel-Canteli lemma will be used at some point, but I could not thought of a good way to connect it to the problem.



Any idea or hint will be really helpful. Thanks.



(edit)



Assume $sum_{ninmathbb{N}}{P(X_n > M)} < infty.$ In fact, as they are i.i.d., $P(X_n > M) = P(X_1>M)$ for all $n.$ Thus $P(X_1>M) = 0$ and $P(sup_{ninmathbb{N}}X_n > M) le sum_{ninmathbb{N}}{P(X_n > M)} = 0.$



Conversely, I've realized that ${sup_{ninmathbb{N}}{X_n} < infty} = bigcup_{m=1}^{infty}bigcap_{n=1}^{infty}{X_n le m}.$






probability-theory measure-theory random-variables independence borel-cantelli-lemmas







share|cite|improve this question











$endgroup$




I am proving the following :



Suppose ${X_n : ninmathbb{N}}$ are i.i.d. random variables. Then $P(sup_{ninmathbb{N}}X_n < infty) = 1$ if and only if $ sum_{ninmathbb{N}}{P(X_n > M)} < infty$ for some $M>0.$



I guess by the form of the problem, that Borel-Canteli lemma will be used at some point, but I could not thought of a good way to connect it to the problem.



Any idea or hint will be really helpful. Thanks.



(edit)



Assume $sum_{ninmathbb{N}}{P(X_n > M)} < infty.$ In fact, as they are i.i.d., $P(X_n > M) = P(X_1>M)$ for all $n.$ Thus $P(X_1>M) = 0$ and $P(sup_{ninmathbb{N}}X_n > M) le sum_{ninmathbb{N}}{P(X_n > M)} = 0.$



Conversely, I've realized that ${sup_{ninmathbb{N}}{X_n} < infty} = bigcup_{m=1}^{infty}bigcap_{n=1}^{infty}{X_n le m}.$






probability-theory measure-theory random-variables independence borel-cantelli-lemmas




probability-theory measure-theory random-variables independence borel-cantelli-lemmas






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 10 '18 at 15:10







Euduardo

















asked Dec 10 '18 at 14:15









EuduardoEuduardo

1638




1638








  • 1




    $begingroup$
    Are you sure the random variables $(X_n)$ are supposed to be i.i.d., or only independent?
    $endgroup$
    – Did
    Dec 10 '18 at 15:20










  • $begingroup$
    I am 100 % sure that they are supposed to be i.i.d., as it was exam problem.
    $endgroup$
    – Euduardo
    Dec 10 '18 at 15:22










  • $begingroup$
    Well then the exam is slightly absurd, but why not.
    $endgroup$
    – Did
    Dec 10 '18 at 15:23










  • $begingroup$
    You're right Did. Now I understand independence is enough, thanks!
    $endgroup$
    – Euduardo
    Dec 11 '18 at 5:09














  • 1




    $begingroup$
    Are you sure the random variables $(X_n)$ are supposed to be i.i.d., or only independent?
    $endgroup$
    – Did
    Dec 10 '18 at 15:20










  • $begingroup$
    I am 100 % sure that they are supposed to be i.i.d., as it was exam problem.
    $endgroup$
    – Euduardo
    Dec 10 '18 at 15:22










  • $begingroup$
    Well then the exam is slightly absurd, but why not.
    $endgroup$
    – Did
    Dec 10 '18 at 15:23










  • $begingroup$
    You're right Did. Now I understand independence is enough, thanks!
    $endgroup$
    – Euduardo
    Dec 11 '18 at 5:09








1




1




$begingroup$
Are you sure the random variables $(X_n)$ are supposed to be i.i.d., or only independent?
$endgroup$
– Did
Dec 10 '18 at 15:20




$begingroup$
Are you sure the random variables $(X_n)$ are supposed to be i.i.d., or only independent?
$endgroup$
– Did
Dec 10 '18 at 15:20












$begingroup$
I am 100 % sure that they are supposed to be i.i.d., as it was exam problem.
$endgroup$
– Euduardo
Dec 10 '18 at 15:22




$begingroup$
I am 100 % sure that they are supposed to be i.i.d., as it was exam problem.
$endgroup$
– Euduardo
Dec 10 '18 at 15:22












$begingroup$
Well then the exam is slightly absurd, but why not.
$endgroup$
– Did
Dec 10 '18 at 15:23




$begingroup$
Well then the exam is slightly absurd, but why not.
$endgroup$
– Did
Dec 10 '18 at 15:23












$begingroup$
You're right Did. Now I understand independence is enough, thanks!
$endgroup$
– Euduardo
Dec 11 '18 at 5:09




$begingroup$
You're right Did. Now I understand independence is enough, thanks!
$endgroup$
– Euduardo
Dec 11 '18 at 5:09










1 Answer
1






active

oldest

votes


















3












$begingroup$

Suppose $P(sup_n X_n < infty) = 1$. Since ${sup_n X_n < infty} = bigcup_{M=1}^infty {sup_n X_n le M}$, we have $lim_{M to infty} P(sup_n X_n le M) = 1$.
But if $P(X_i > M) > 0$, $P(sup_n X_n > M) = 1$. So for some $M$ we must have $P(X_i > M) = 0$.



Conversely, if $P(sup_n X_n < infty) < 1$, then for all $M$ we have $P(sup_n X_n > M) > 0$,
and $P(X_i > M) > 0$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    How do you go from $P[X_i >M] >0$ to $P[sup_n X_n >M] = 1$?
    $endgroup$
    – copper.hat
    Dec 10 '18 at 17:07










  • $begingroup$
    Second Borel-Cantelli lemma.
    $endgroup$
    – Robert Israel
    Dec 10 '18 at 18:21










  • $begingroup$
    Using the fact that $X_n$ are iid, I see. Thanks.
    $endgroup$
    – copper.hat
    Dec 10 '18 at 18:23











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%2f3033962%2fequivalence-of-the-condition-that-the-supremum-of-i-i-d-rvs-are-finite-a-s%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









3












$begingroup$

Suppose $P(sup_n X_n < infty) = 1$. Since ${sup_n X_n < infty} = bigcup_{M=1}^infty {sup_n X_n le M}$, we have $lim_{M to infty} P(sup_n X_n le M) = 1$.
But if $P(X_i > M) > 0$, $P(sup_n X_n > M) = 1$. So for some $M$ we must have $P(X_i > M) = 0$.



Conversely, if $P(sup_n X_n < infty) < 1$, then for all $M$ we have $P(sup_n X_n > M) > 0$,
and $P(X_i > M) > 0$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    How do you go from $P[X_i >M] >0$ to $P[sup_n X_n >M] = 1$?
    $endgroup$
    – copper.hat
    Dec 10 '18 at 17:07










  • $begingroup$
    Second Borel-Cantelli lemma.
    $endgroup$
    – Robert Israel
    Dec 10 '18 at 18:21










  • $begingroup$
    Using the fact that $X_n$ are iid, I see. Thanks.
    $endgroup$
    – copper.hat
    Dec 10 '18 at 18:23
















3












$begingroup$

Suppose $P(sup_n X_n < infty) = 1$. Since ${sup_n X_n < infty} = bigcup_{M=1}^infty {sup_n X_n le M}$, we have $lim_{M to infty} P(sup_n X_n le M) = 1$.
But if $P(X_i > M) > 0$, $P(sup_n X_n > M) = 1$. So for some $M$ we must have $P(X_i > M) = 0$.



Conversely, if $P(sup_n X_n < infty) < 1$, then for all $M$ we have $P(sup_n X_n > M) > 0$,
and $P(X_i > M) > 0$.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    How do you go from $P[X_i >M] >0$ to $P[sup_n X_n >M] = 1$?
    $endgroup$
    – copper.hat
    Dec 10 '18 at 17:07










  • $begingroup$
    Second Borel-Cantelli lemma.
    $endgroup$
    – Robert Israel
    Dec 10 '18 at 18:21










  • $begingroup$
    Using the fact that $X_n$ are iid, I see. Thanks.
    $endgroup$
    – copper.hat
    Dec 10 '18 at 18:23














3












3








3





$begingroup$

Suppose $P(sup_n X_n < infty) = 1$. Since ${sup_n X_n < infty} = bigcup_{M=1}^infty {sup_n X_n le M}$, we have $lim_{M to infty} P(sup_n X_n le M) = 1$.
But if $P(X_i > M) > 0$, $P(sup_n X_n > M) = 1$. So for some $M$ we must have $P(X_i > M) = 0$.



Conversely, if $P(sup_n X_n < infty) < 1$, then for all $M$ we have $P(sup_n X_n > M) > 0$,
and $P(X_i > M) > 0$.






share|cite|improve this answer









$endgroup$



Suppose $P(sup_n X_n < infty) = 1$. Since ${sup_n X_n < infty} = bigcup_{M=1}^infty {sup_n X_n le M}$, we have $lim_{M to infty} P(sup_n X_n le M) = 1$.
But if $P(X_i > M) > 0$, $P(sup_n X_n > M) = 1$. So for some $M$ we must have $P(X_i > M) = 0$.



Conversely, if $P(sup_n X_n < infty) < 1$, then for all $M$ we have $P(sup_n X_n > M) > 0$,
and $P(X_i > M) > 0$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 10 '18 at 15:11









Robert IsraelRobert Israel

324k23214468




324k23214468












  • $begingroup$
    How do you go from $P[X_i >M] >0$ to $P[sup_n X_n >M] = 1$?
    $endgroup$
    – copper.hat
    Dec 10 '18 at 17:07










  • $begingroup$
    Second Borel-Cantelli lemma.
    $endgroup$
    – Robert Israel
    Dec 10 '18 at 18:21










  • $begingroup$
    Using the fact that $X_n$ are iid, I see. Thanks.
    $endgroup$
    – copper.hat
    Dec 10 '18 at 18:23


















  • $begingroup$
    How do you go from $P[X_i >M] >0$ to $P[sup_n X_n >M] = 1$?
    $endgroup$
    – copper.hat
    Dec 10 '18 at 17:07










  • $begingroup$
    Second Borel-Cantelli lemma.
    $endgroup$
    – Robert Israel
    Dec 10 '18 at 18:21










  • $begingroup$
    Using the fact that $X_n$ are iid, I see. Thanks.
    $endgroup$
    – copper.hat
    Dec 10 '18 at 18:23
















$begingroup$
How do you go from $P[X_i >M] >0$ to $P[sup_n X_n >M] = 1$?
$endgroup$
– copper.hat
Dec 10 '18 at 17:07




$begingroup$
How do you go from $P[X_i >M] >0$ to $P[sup_n X_n >M] = 1$?
$endgroup$
– copper.hat
Dec 10 '18 at 17:07












$begingroup$
Second Borel-Cantelli lemma.
$endgroup$
– Robert Israel
Dec 10 '18 at 18:21




$begingroup$
Second Borel-Cantelli lemma.
$endgroup$
– Robert Israel
Dec 10 '18 at 18:21












$begingroup$
Using the fact that $X_n$ are iid, I see. Thanks.
$endgroup$
– copper.hat
Dec 10 '18 at 18:23




$begingroup$
Using the fact that $X_n$ are iid, I see. Thanks.
$endgroup$
– copper.hat
Dec 10 '18 at 18:23


















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%2f3033962%2fequivalence-of-the-condition-that-the-supremum-of-i-i-d-rvs-are-finite-a-s%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

Plaza Victoria

Image
Plaza Victoria Plaza Victoria en 2017 Otros nombres Plaza de La Victoria Localización El Almendral, Valparaíso, Chile Vías adyacentes Chacabuco, Molina, Plaza Victoria y Edwards Creación 1841 Metro Bellavista Ubicación 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 Coordenadas: 33°02′46″S 71°37′11″O  /  -33.04624167, -71.61980833 [editar datos en Wikidata] La Plaza Victoria , también llamada Plaza de la Victoria , [ 1 ] ​ es una plaza ubicada en el sector El Almendral del plan de la ciudad de Valparaíso, Chile, de importancia patrimonial, que incluye valiosas estatuas y especies vegetales. [ 2 ] ​ La plaza se creó como tal en la primera mitad del siglo XIX, tomando previamente los nombres de Plaza Nueva y Plaza de Orrego . Si bien desde sus inicios tuvo relevancia histórica, no fue sino a partir de fines de los años 1860 cuando comenzó a cobrar mayor importancia y se convirtió en uno de los principales centros sociales de l
Read more

Puebla de Zaragoza

Image
Para otros usos de este término, véase Puebla (desambiguación). Heroica Puebla de Zaragoza Localidad De arriba a abajo de izquierda a derecha: Estrella de Puebla, Volcán Popocatépetl, Catedral de Puebla, Monumento a Ignacio Zaragoza, Estadio Cuauhtémoc, Angelópolis y Vista panorámica de la ciudad. Escudo Otros nombres: El relicario de América/ La ciudad de los Ángeles/ La Angelópolis Lema: Muy noble y muy leal ciudad de Puebla de los Ángeles Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en México Heroica Puebla de Zaragoza Localización de Heroica Puebla de Zaragoza en Puebla Mapa interactivo Coordenadas 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Coordenadas: 19°03′05″N 98°13′04″O  /  19.051388888889, -98.217777777778 Idioma oficial Español Entidad Localidad  • País   México  • Estado Puebla  • Municipio Puebla Presidente municipal Claudia Rivera Vivanc
Read more

Musa

Image
Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo
Read more