Show that $sum_{k=1}^{infty} sigma^2_k <infty$ implies $|sum_{k=1}^{infty} X_k|<infty $ almost surely.
$begingroup$
Suppose that $X_1, X_2, X_3,ldots$ are sequence of independent random variables such that $mu_k= 0$ and $ sigma^2_k =operatorname{Var}(X_k)< infty$ for all $k$. Then
show that $sum_{k=1}^{infty} sigma^2_k <infty$ implies $|sum_{k=1}^{infty} X_k|<infty $ almost surely.
I was wondering how could I use the facts: 1) if a martingale $M$ is bounded in $L^2 $ then $lim M_n$ exists almost surely.
2) Orthogonality of increments of $M$ to prove the above statement. I would like to see the solution in explained way.
sequences-and-series probability-theory random-variables martingales variance
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
$endgroup$
|
show 2 more comments
$begingroup$
Suppose that $X_1, X_2, X_3,ldots$ are sequence of independent random variables such that $mu_k= 0$ and $ sigma^2_k =operatorname{Var}(X_k)< infty$ for all $k$. Then
show that $sum_{k=1}^{infty} sigma^2_k <infty$ implies $|sum_{k=1}^{infty} X_k|<infty $ almost surely.
I was wondering how could I use the facts: 1) if a martingale $M$ is bounded in $L^2 $ then $lim M_n$ exists almost surely.
2) Orthogonality of increments of $M$ to prove the above statement. I would like to see the solution in explained way.
sequences-and-series probability-theory random-variables martingales variance
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
$endgroup$
$begingroup$
If $P(|Z| ge b) ge a$ then $Var(Z) ge a b^2$. Using that $Var(Y_n) < C$ and $Var(Y-Y_n) to 0$ show that the sequence of random variables $Y_n = sum_{k=1}^n X_k$ is uniformly bounded almost surely, and that it converges almost surely.
$endgroup$
– reuns
Apr 7 '17 at 0:15
$begingroup$
Then where are the orthogonality applied?
$endgroup$
– 00012 suxn
Apr 7 '17 at 11:56
$begingroup$
$Var(Y_n) = sum_{k=1}^n Var(X_k)$
$endgroup$
– reuns
Apr 7 '17 at 11:59
$begingroup$
Thank you, Do you have some idea on the following one: math.stackexchange.com/questions/2221753/…
$endgroup$
– 00012 suxn
Apr 7 '17 at 12:11
$begingroup$
Did you complete this question ? Write your own answer then.
$endgroup$
– reuns
Apr 7 '17 at 12:12
|
show 2 more comments
$begingroup$
Suppose that $X_1, X_2, X_3,ldots$ are sequence of independent random variables such that $mu_k= 0$ and $ sigma^2_k =operatorname{Var}(X_k)< infty$ for all $k$. Then
show that $sum_{k=1}^{infty} sigma^2_k <infty$ implies $|sum_{k=1}^{infty} X_k|<infty $ almost surely.
I was wondering how could I use the facts: 1) if a martingale $M$ is bounded in $L^2 $ then $lim M_n$ exists almost surely.
2) Orthogonality of increments of $M$ to prove the above statement. I would like to see the solution in explained way.
sequences-and-series probability-theory random-variables martingales variance
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
$endgroup$
Suppose that $X_1, X_2, X_3,ldots$ are sequence of independent random variables such that $mu_k= 0$ and $ sigma^2_k =operatorname{Var}(X_k)< infty$ for all $k$. Then
show that $sum_{k=1}^{infty} sigma^2_k <infty$ implies $|sum_{k=1}^{infty} X_k|<infty $ almost surely.
I was wondering how could I use the facts: 1) if a martingale $M$ is bounded in $L^2 $ then $lim M_n$ exists almost surely.
2) Orthogonality of increments of $M$ to prove the above statement. I would like to see the solution in explained way.
sequences-and-series probability-theory random-variables martingales variance
sequences-and-series probability-theory random-variables martingales variance
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
edited Nov 26 '18 at 22:23
Davide Giraudo
125k16150261
125k16150261
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
asked Apr 6 '17 at 23:51
00012 suxn00012 suxn
39919
39919
$begingroup$
If $P(|Z| ge b) ge a$ then $Var(Z) ge a b^2$. Using that $Var(Y_n) < C$ and $Var(Y-Y_n) to 0$ show that the sequence of random variables $Y_n = sum_{k=1}^n X_k$ is uniformly bounded almost surely, and that it converges almost surely.
$endgroup$
– reuns
Apr 7 '17 at 0:15
$begingroup$
Then where are the orthogonality applied?
$endgroup$
– 00012 suxn
Apr 7 '17 at 11:56
$begingroup$
$Var(Y_n) = sum_{k=1}^n Var(X_k)$
$endgroup$
– reuns
Apr 7 '17 at 11:59
$begingroup$
Thank you, Do you have some idea on the following one: math.stackexchange.com/questions/2221753/…
$endgroup$
– 00012 suxn
Apr 7 '17 at 12:11
$begingroup$
Did you complete this question ? Write your own answer then.
$endgroup$
– reuns
Apr 7 '17 at 12:12
|
show 2 more comments
$begingroup$
If $P(|Z| ge b) ge a$ then $Var(Z) ge a b^2$. Using that $Var(Y_n) < C$ and $Var(Y-Y_n) to 0$ show that the sequence of random variables $Y_n = sum_{k=1}^n X_k$ is uniformly bounded almost surely, and that it converges almost surely.
$endgroup$
– reuns
Apr 7 '17 at 0:15
$begingroup$
Then where are the orthogonality applied?
$endgroup$
– 00012 suxn
Apr 7 '17 at 11:56
$begingroup$
$Var(Y_n) = sum_{k=1}^n Var(X_k)$
$endgroup$
– reuns
Apr 7 '17 at 11:59
$begingroup$
Thank you, Do you have some idea on the following one: math.stackexchange.com/questions/2221753/…
$endgroup$
– 00012 suxn
Apr 7 '17 at 12:11
$begingroup$
Did you complete this question ? Write your own answer then.
$endgroup$
– reuns
Apr 7 '17 at 12:12
$begingroup$
If $P(|Z| ge b) ge a$ then $Var(Z) ge a b^2$. Using that $Var(Y_n) < C$ and $Var(Y-Y_n) to 0$ show that the sequence of random variables $Y_n = sum_{k=1}^n X_k$ is uniformly bounded almost surely, and that it converges almost surely.
$endgroup$
– reuns
Apr 7 '17 at 0:15
$begingroup$
If $P(|Z| ge b) ge a$ then $Var(Z) ge a b^2$. Using that $Var(Y_n) < C$ and $Var(Y-Y_n) to 0$ show that the sequence of random variables $Y_n = sum_{k=1}^n X_k$ is uniformly bounded almost surely, and that it converges almost surely.
$endgroup$
– reuns
Apr 7 '17 at 0:15
$begingroup$
Then where are the orthogonality applied?
$endgroup$
– 00012 suxn
Apr 7 '17 at 11:56
$begingroup$
Then where are the orthogonality applied?
$endgroup$
– 00012 suxn
Apr 7 '17 at 11:56
$begingroup$
$Var(Y_n) = sum_{k=1}^n Var(X_k)$
$endgroup$
– reuns
Apr 7 '17 at 11:59
$begingroup$
$Var(Y_n) = sum_{k=1}^n Var(X_k)$
$endgroup$
– reuns
Apr 7 '17 at 11:59
$begingroup$
Thank you, Do you have some idea on the following one: math.stackexchange.com/questions/2221753/…
$endgroup$
– 00012 suxn
Apr 7 '17 at 12:11
$begingroup$
Thank you, Do you have some idea on the following one: math.stackexchange.com/questions/2221753/…
$endgroup$
– 00012 suxn
Apr 7 '17 at 12:11
$begingroup$
Did you complete this question ? Write your own answer then.
$endgroup$
– reuns
Apr 7 '17 at 12:12
$begingroup$
Did you complete this question ? Write your own answer then.
$endgroup$
– reuns
Apr 7 '17 at 12:12
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Let $M_n:=sum_{i=1}^nX_i$. Defining $mathcal F_n$ as the $sigma$-algebra generated by $X_i$, $1leqslant ileqslant n$. Then $(S_n,mathcal F_n)$ is a martingale and
using orthogonality of increments,
$$
mathbb Eleft[M_n^2right]=sum_{k=1}^nmathbb Eleft[X_k^2right]=sum_{k=1}^nsigma_k^2
$$
hence
$$
sup_nmathbb Eleft[M_n^2right]leqslant sum_{k=1}^{+infty}sigma_k^2.
$$
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Let $M_n:=sum_{i=1}^nX_i$. Defining $mathcal F_n$ as the $sigma$-algebra generated by $X_i$, $1leqslant ileqslant n$. Then $(S_n,mathcal F_n)$ is a martingale and
using orthogonality of increments,
$$
mathbb Eleft[M_n^2right]=sum_{k=1}^nmathbb Eleft[X_k^2right]=sum_{k=1}^nsigma_k^2
$$
hence
$$
sup_nmathbb Eleft[M_n^2right]leqslant sum_{k=1}^{+infty}sigma_k^2.
$$
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
add a comment |
$begingroup$
Let $M_n:=sum_{i=1}^nX_i$. Defining $mathcal F_n$ as the $sigma$-algebra generated by $X_i$, $1leqslant ileqslant n$. Then $(S_n,mathcal F_n)$ is a martingale and
using orthogonality of increments,
$$
mathbb Eleft[M_n^2right]=sum_{k=1}^nmathbb Eleft[X_k^2right]=sum_{k=1}^nsigma_k^2
$$
hence
$$
sup_nmathbb Eleft[M_n^2right]leqslant sum_{k=1}^{+infty}sigma_k^2.
$$
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
add a comment |
$begingroup$
Let $M_n:=sum_{i=1}^nX_i$. Defining $mathcal F_n$ as the $sigma$-algebra generated by $X_i$, $1leqslant ileqslant n$. Then $(S_n,mathcal F_n)$ is a martingale and
using orthogonality of increments,
$$
mathbb Eleft[M_n^2right]=sum_{k=1}^nmathbb Eleft[X_k^2right]=sum_{k=1}^nsigma_k^2
$$
hence
$$
sup_nmathbb Eleft[M_n^2right]leqslant sum_{k=1}^{+infty}sigma_k^2.
$$
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
$endgroup$
Let $M_n:=sum_{i=1}^nX_i$. Defining $mathcal F_n$ as the $sigma$-algebra generated by $X_i$, $1leqslant ileqslant n$. Then $(S_n,mathcal F_n)$ is a martingale and
using orthogonality of increments,
$$
mathbb Eleft[M_n^2right]=sum_{k=1}^nmathbb Eleft[X_k^2right]=sum_{k=1}^nsigma_k^2
$$
hence
$$
sup_nmathbb Eleft[M_n^2right]leqslant sum_{k=1}^{+infty}sigma_k^2.
$$
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
answered Nov 24 '18 at 22:14
Davide GiraudoDavide Giraudo
125k16150261
125k16150261
add a comment |
add a comment |
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$begingroup$
If $P(|Z| ge b) ge a$ then $Var(Z) ge a b^2$. Using that $Var(Y_n) < C$ and $Var(Y-Y_n) to 0$ show that the sequence of random variables $Y_n = sum_{k=1}^n X_k$ is uniformly bounded almost surely, and that it converges almost surely.
$endgroup$
– reuns
Apr 7 '17 at 0:15
$begingroup$
Then where are the orthogonality applied?
$endgroup$
– 00012 suxn
Apr 7 '17 at 11:56
$begingroup$
$Var(Y_n) = sum_{k=1}^n Var(X_k)$
$endgroup$
– reuns
Apr 7 '17 at 11:59
$begingroup$
Thank you, Do you have some idea on the following one: math.stackexchange.com/questions/2221753/…
$endgroup$
– 00012 suxn
Apr 7 '17 at 12:11
$begingroup$
Did you complete this question ? Write your own answer then.
$endgroup$
– reuns
Apr 7 '17 at 12:12