Mixing
Expectation: Is this statment true or is there a counter example [closed]
Clash Royale CLAN TAG#URR8PPP
up vote
0
down vote
favorite
Let $(X_i)_iin (1,...,N)$ be a sequence of i.i.d random variables with $0<mathbbE(X_i^2)<0$. Is
$$sup_i|X_i|<infty$$
a consequence of the strong law of large numbers, or is there a counterexample?
random-variables examples-counterexamples law-of-large-numbers
asked Jul 31 at 16:11
Dai Jinaid
194
closed as unclear what you're asking by Clement C., amWhy, Isaac Browne, Xander Henderson, max_zorn Aug 1 at 5:52
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |Â
up vote
0
down vote
favorite
Let $(X_i)_iin (1,...,N)$ be a sequence of i.i.d random variables with $0<mathbbE(X_i^2)<0$. Is
$$sup_i|X_i|<infty$$
a consequence of the strong law of large numbers, or is there a counterexample?
random-variables examples-counterexamples law-of-large-numbers
asked Jul 31 at 16:11
Dai Jinaid
194
closed as unclear what you're asking by Clement C., amWhy, Isaac Browne, Xander Henderson, max_zorn Aug 1 at 5:52
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
I don't understand what the displayed inequality means. There are finitely many variables; thus their supremum is their maximum and is necessarily finite.
– joriki
Jul 31 at 16:18
5
$0<mathbbE(X_i^2)<0$ looks meaningless to me.
– Arthur
Jul 31 at 16:23
I'm guessing you meant $0 < operatorname E(X_i^2) <+infty. qquad$
– Michael Hardy
Jul 31 at 18:20
Where you wrote $0<operatorname E(X_i^2)<0,$ I will guess that you meant $0<operatorname E(X_i^2)<+infty.$ You wrote $(X_i)_iin(1,ldots,N),$ which, taken literally, means you have only finitely many random variables. Also, not that one should write $iin1,ldots,N,$ with $textcurly braces$ indicating a set rather than a tuple. Since the strong law of large numbers applies only to an infinite sequence, not to a finite sequence, I will guess that you actually intended an infinite sequence. You could have written $(X_i)_i,in,mathbb N$ or $(X_i)_i,=,1,2,3,ldots.quad$
– Michael Hardy
Jul 31 at 19:47
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $(X_i)_iin (1,...,N)$ be a sequence of i.i.d random variables with $0<mathbbE(X_i^2)<0$. Is
$$sup_i|X_i|<infty$$
a consequence of the strong law of large numbers, or is there a counterexample?
random-variables examples-counterexamples law-of-large-numbers
asked Jul 31 at 16:11
Dai Jinaid
194
Let $(X_i)_iin (1,...,N)$ be a sequence of i.i.d random variables with $0<mathbbE(X_i^2)<0$. Is
$$sup_i|X_i|<infty$$
a consequence of the strong law of large numbers, or is there a counterexample?
random-variables examples-counterexamples law-of-large-numbers
asked Jul 31 at 16:11
Dai Jinaid
194
asked Jul 31 at 16:11
Dai Jinaid
194
asked Jul 31 at 16:11
Dai Jinaid
194
asked Jul 31 at 16:11
Dai Jinaid
194
194
closed as unclear what you're asking by Clement C., amWhy, Isaac Browne, Xander Henderson, max_zorn Aug 1 at 5:52
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
closed as unclear what you're asking by Clement C., amWhy, Isaac Browne, Xander Henderson, max_zorn Aug 1 at 5:52
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
2
I don't understand what the displayed inequality means. There are finitely many variables; thus their supremum is their maximum and is necessarily finite.
– joriki
Jul 31 at 16:18
5
$0<mathbbE(X_i^2)<0$ looks meaningless to me.
– Arthur
Jul 31 at 16:23
I'm guessing you meant $0 < operatorname E(X_i^2) <+infty. qquad$
– Michael Hardy
Jul 31 at 18:20
Where you wrote $0<operatorname E(X_i^2)<0,$ I will guess that you meant $0<operatorname E(X_i^2)<+infty.$ You wrote $(X_i)_iin(1,ldots,N),$ which, taken literally, means you have only finitely many random variables. Also, not that one should write $iin1,ldots,N,$ with $textcurly braces$ indicating a set rather than a tuple. Since the strong law of large numbers applies only to an infinite sequence, not to a finite sequence, I will guess that you actually intended an infinite sequence. You could have written $(X_i)_i,in,mathbb N$ or $(X_i)_i,=,1,2,3,ldots.quad$
– Michael Hardy
Jul 31 at 19:47
add a comment |Â
2
I don't understand what the displayed inequality means. There are finitely many variables; thus their supremum is their maximum and is necessarily finite.
– joriki
Jul 31 at 16:18
5
$0<mathbbE(X_i^2)<0$ looks meaningless to me.
– Arthur
Jul 31 at 16:23
I'm guessing you meant $0 < operatorname E(X_i^2) <+infty. qquad$
– Michael Hardy
Jul 31 at 18:20
Where you wrote $0<operatorname E(X_i^2)<0,$ I will guess that you meant $0<operatorname E(X_i^2)<+infty.$ You wrote $(X_i)_iin(1,ldots,N),$ which, taken literally, means you have only finitely many random variables. Also, not that one should write $iin1,ldots,N,$ with $textcurly braces$ indicating a set rather than a tuple. Since the strong law of large numbers applies only to an infinite sequence, not to a finite sequence, I will guess that you actually intended an infinite sequence. You could have written $(X_i)_i,in,mathbb N$ or $(X_i)_i,=,1,2,3,ldots.quad$
– Michael Hardy
Jul 31 at 19:47
2
2
I don't understand what the displayed inequality means. There are finitely many variables; thus their supremum is their maximum and is necessarily finite.
– joriki
Jul 31 at 16:18
I don't understand what the displayed inequality means. There are finitely many variables; thus their supremum is their maximum and is necessarily finite.
– joriki
Jul 31 at 16:18
5
5
$0<mathbbE(X_i^2)<0$ looks meaningless to me.
– Arthur
Jul 31 at 16:23
$0<mathbbE(X_i^2)<0$ looks meaningless to me.
– Arthur
Jul 31 at 16:23
I'm guessing you meant $0 < operatorname E(X_i^2) <+infty. qquad$
– Michael Hardy
Jul 31 at 18:20
I'm guessing you meant $0 < operatorname E(X_i^2) <+infty. qquad$
– Michael Hardy
Jul 31 at 18:20
Where you wrote $0<operatorname E(X_i^2)<0,$ I will guess that you meant $0<operatorname E(X_i^2)<+infty.$ You wrote $(X_i)_iin(1,ldots,N),$ which, taken literally, means you have only finitely many random variables. Also, not that one should write $iin1,ldots,N,$ with $textcurly braces$ indicating a set rather than a tuple. Since the strong law of large numbers applies only to an infinite sequence, not to a finite sequence, I will guess that you actually intended an infinite sequence. You could have written $(X_i)_i,in,mathbb N$ or $(X_i)_i,=,1,2,3,ldots.quad$
– Michael Hardy
Jul 31 at 19:47
Where you wrote $0<operatorname E(X_i^2)<0,$ I will guess that you meant $0<operatorname E(X_i^2)<+infty.$ You wrote $(X_i)_iin(1,ldots,N),$ which, taken literally, means you have only finitely many random variables. Also, not that one should write $iin1,ldots,N,$ with $textcurly braces$ indicating a set rather than a tuple. Since the strong law of large numbers applies only to an infinite sequence, not to a finite sequence, I will guess that you actually intended an infinite sequence. You could have written $(X_i)_i,in,mathbb N$ or $(X_i)_i,=,1,2,3,ldots.quad$
– Michael Hardy
Jul 31 at 19:47
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
I assume you are asking about an infinite sequence $X_1,X_2,dots$ of iid random variables, and you want to know whether $sup_i |X_i|<infty$ holds almost surely.
If $X_1$ is bounded, meaning $P(|X_1|le M)=1$ , then trivially we will have $P(sup |X_i|le M)=1$.
If $X_1$ is unbounded, however, then the answer is no. For any $M>0$, then the event $X_i$ will have a nonzero probability, so with probability one it will occur for some $i$. Therefore, with probability one, $sup_i |X_i|ge M$ for all positive integers $M$, so $sup_i |X_i| = infty$ with probability $1$.
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
I assume you are asking about an infinite sequence $X_1,X_2,dots$ of iid random variables, and you want to know whether $sup_i |X_i|<infty$ holds almost surely.
If $X_1$ is bounded, meaning $P(|X_1|le M)=1$ , then trivially we will have $P(sup |X_i|le M)=1$.
If $X_1$ is unbounded, however, then the answer is no. For any $M>0$, then the event $X_i$ will have a nonzero probability, so with probability one it will occur for some $i$. Therefore, with probability one, $sup_i |X_i|ge M$ for all positive integers $M$, so $sup_i |X_i| = infty$ with probability $1$.
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
add a comment |Â
up vote
2
down vote
I assume you are asking about an infinite sequence $X_1,X_2,dots$ of iid random variables, and you want to know whether $sup_i |X_i|<infty$ holds almost surely.
If $X_1$ is bounded, meaning $P(|X_1|le M)=1$ , then trivially we will have $P(sup |X_i|le M)=1$.
If $X_1$ is unbounded, however, then the answer is no. For any $M>0$, then the event $X_i$ will have a nonzero probability, so with probability one it will occur for some $i$. Therefore, with probability one, $sup_i |X_i|ge M$ for all positive integers $M$, so $sup_i |X_i| = infty$ with probability $1$.
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
add a comment |Â
up vote
2
down vote
up vote
2
down vote
I assume you are asking about an infinite sequence $X_1,X_2,dots$ of iid random variables, and you want to know whether $sup_i |X_i|<infty$ holds almost surely.
If $X_1$ is bounded, meaning $P(|X_1|le M)=1$ , then trivially we will have $P(sup |X_i|le M)=1$.
If $X_1$ is unbounded, however, then the answer is no. For any $M>0$, then the event $X_i$ will have a nonzero probability, so with probability one it will occur for some $i$. Therefore, with probability one, $sup_i |X_i|ge M$ for all positive integers $M$, so $sup_i |X_i| = infty$ with probability $1$.
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
I assume you are asking about an infinite sequence $X_1,X_2,dots$ of iid random variables, and you want to know whether $sup_i |X_i|<infty$ holds almost surely.
If $X_1$ is bounded, meaning $P(|X_1|le M)=1$ , then trivially we will have $P(sup |X_i|le M)=1$.
If $X_1$ is unbounded, however, then the answer is no. For any $M>0$, then the event $X_i$ will have a nonzero probability, so with probability one it will occur for some $i$. Therefore, with probability one, $sup_i |X_i|ge M$ for all positive integers $M$, so $sup_i |X_i| = infty$ with probability $1$.
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
answered Jul 31 at 19:29
Mike Earnest
14.7k11644
14.7k11644
add a comment |Â
add a comment |Â
2
I don't understand what the displayed inequality means. There are finitely many variables; thus their supremum is their maximum and is necessarily finite.
– joriki
Jul 31 at 16:18
5
$0<mathbbE(X_i^2)<0$ looks meaningless to me.
– Arthur
Jul 31 at 16:23
I'm guessing you meant $0 < operatorname E(X_i^2) <+infty. qquad$
– Michael Hardy
Jul 31 at 18:20
Where you wrote $0<operatorname E(X_i^2)<0,$ I will guess that you meant $0<operatorname E(X_i^2)<+infty.$ You wrote $(X_i)_iin(1,ldots,N),$ which, taken literally, means you have only finitely many random variables. Also, not that one should write $iin1,ldots,N,$ with $textcurly braces$ indicating a set rather than a tuple. Since the strong law of large numbers applies only to an infinite sequence, not to a finite sequence, I will guess that you actually intended an infinite sequence. You could have written $(X_i)_i,in,mathbb N$ or $(X_i)_i,=,1,2,3,ldots.quad$
– Michael Hardy
Jul 31 at 19:47