Reference request: Strong Law of Large Numbers for V-statistics
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I'm requesting a reference for a Strong Law of Large Numbers theorem for V-statistics (similar to Hoeffding's 1961 paper for U-statistics). That is, I am searching for an almost sure convergence theorem, saying
$$
V_n = frac{1}{n^k}sum_{i_1=1}^n cdots sum_{i_k=1}^n hleft(X_{i_1},...,X_{i_k}right) stackrel{mbox{a.s.}}{longrightarrow}_n quad?
$$
where $(X_n)$ is an i.i.d. sequence of random elements with values in some space $mathcal{X}$ and $h:mathcal{X}^k to mathbb{R}$ is a symmetric kernel with some moment requirements.
If Hoeffdings SSLN for U-statistics can be used to derive this result, an explanation of how this is done, would also more than suffice.
probability-theory statistics reference-request asymptotics law-of-large-numbers
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
$endgroup$
add a comment |
$begingroup$
I'm requesting a reference for a Strong Law of Large Numbers theorem for V-statistics (similar to Hoeffding's 1961 paper for U-statistics). That is, I am searching for an almost sure convergence theorem, saying
$$
V_n = frac{1}{n^k}sum_{i_1=1}^n cdots sum_{i_k=1}^n hleft(X_{i_1},...,X_{i_k}right) stackrel{mbox{a.s.}}{longrightarrow}_n quad?
$$
where $(X_n)$ is an i.i.d. sequence of random elements with values in some space $mathcal{X}$ and $h:mathcal{X}^k to mathbb{R}$ is a symmetric kernel with some moment requirements.
If Hoeffdings SSLN for U-statistics can be used to derive this result, an explanation of how this is done, would also more than suffice.
probability-theory statistics reference-request asymptotics law-of-large-numbers
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
$endgroup$
$begingroup$
They are equivalent. Explanation: $n^kV_n=n(n-1)cdots(n-k+1)U_n+R_n$ where the rest $R_n$ is the sum over every $k$uple with at least two indices equal. There are $O(n^{k-1})$ terms in $R_n$ hence $n^{-k}R_nto0$ almost surely (under various moment hypotheses) hence you are done.
$endgroup$
– Did
Aug 30 '16 at 12:34
$begingroup$
Thanks Did, much appreciated.
$endgroup$
– Martin
Aug 30 '16 at 12:53
add a comment |
$begingroup$
I'm requesting a reference for a Strong Law of Large Numbers theorem for V-statistics (similar to Hoeffding's 1961 paper for U-statistics). That is, I am searching for an almost sure convergence theorem, saying
$$
V_n = frac{1}{n^k}sum_{i_1=1}^n cdots sum_{i_k=1}^n hleft(X_{i_1},...,X_{i_k}right) stackrel{mbox{a.s.}}{longrightarrow}_n quad?
$$
where $(X_n)$ is an i.i.d. sequence of random elements with values in some space $mathcal{X}$ and $h:mathcal{X}^k to mathbb{R}$ is a symmetric kernel with some moment requirements.
If Hoeffdings SSLN for U-statistics can be used to derive this result, an explanation of how this is done, would also more than suffice.
probability-theory statistics reference-request asymptotics law-of-large-numbers
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
$endgroup$
I'm requesting a reference for a Strong Law of Large Numbers theorem for V-statistics (similar to Hoeffding's 1961 paper for U-statistics). That is, I am searching for an almost sure convergence theorem, saying
$$
V_n = frac{1}{n^k}sum_{i_1=1}^n cdots sum_{i_k=1}^n hleft(X_{i_1},...,X_{i_k}right) stackrel{mbox{a.s.}}{longrightarrow}_n quad?
$$
where $(X_n)$ is an i.i.d. sequence of random elements with values in some space $mathcal{X}$ and $h:mathcal{X}^k to mathbb{R}$ is a symmetric kernel with some moment requirements.
If Hoeffdings SSLN for U-statistics can be used to derive this result, an explanation of how this is done, would also more than suffice.
probability-theory statistics reference-request asymptotics law-of-large-numbers
probability-theory statistics reference-request asymptotics law-of-large-numbers
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
edited Jan 2 at 13:27
Davide Giraudo
128k17155268
128k17155268
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
asked Aug 30 '16 at 12:28
MartinMartin
1,1811019
1,1811019
$begingroup$
They are equivalent. Explanation: $n^kV_n=n(n-1)cdots(n-k+1)U_n+R_n$ where the rest $R_n$ is the sum over every $k$uple with at least two indices equal. There are $O(n^{k-1})$ terms in $R_n$ hence $n^{-k}R_nto0$ almost surely (under various moment hypotheses) hence you are done.
$endgroup$
– Did
Aug 30 '16 at 12:34
$begingroup$
Thanks Did, much appreciated.
$endgroup$
– Martin
Aug 30 '16 at 12:53
add a comment |
$begingroup$
They are equivalent. Explanation: $n^kV_n=n(n-1)cdots(n-k+1)U_n+R_n$ where the rest $R_n$ is the sum over every $k$uple with at least two indices equal. There are $O(n^{k-1})$ terms in $R_n$ hence $n^{-k}R_nto0$ almost surely (under various moment hypotheses) hence you are done.
$endgroup$
– Did
Aug 30 '16 at 12:34
$begingroup$
Thanks Did, much appreciated.
$endgroup$
– Martin
Aug 30 '16 at 12:53
$begingroup$
They are equivalent. Explanation: $n^kV_n=n(n-1)cdots(n-k+1)U_n+R_n$ where the rest $R_n$ is the sum over every $k$uple with at least two indices equal. There are $O(n^{k-1})$ terms in $R_n$ hence $n^{-k}R_nto0$ almost surely (under various moment hypotheses) hence you are done.
$endgroup$
– Did
Aug 30 '16 at 12:34
$begingroup$
They are equivalent. Explanation: $n^kV_n=n(n-1)cdots(n-k+1)U_n+R_n$ where the rest $R_n$ is the sum over every $k$uple with at least two indices equal. There are $O(n^{k-1})$ terms in $R_n$ hence $n^{-k}R_nto0$ almost surely (under various moment hypotheses) hence you are done.
$endgroup$
– Did
Aug 30 '16 at 12:34
$begingroup$
Thanks Did, much appreciated.
$endgroup$
– Martin
Aug 30 '16 at 12:53
$begingroup$
Thanks Did, much appreciated.
$endgroup$
– Martin
Aug 30 '16 at 12:53
add a comment |
1 Answer
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I found four references:
- Giné, Evarist, and Joel Zinn. "Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics." Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Birkhäuser Boston, 1992.
- Kondo, Masao, and Hajime Yamato. "Almost sure convergence of a linear combination of U-statistics." Scientiae Mathematicae japonicae 55.3 (2002): 605-613.
- Borovskikh, Yuri Vasilevich. U-statistics in Banach Spaces. VSP, 1996.
- Korolyuk, Vladimir S., and Yuri V. Borovskich. Theory of U-statistics. Vol. 273. Springer Science & Business Media, 2013.
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
$endgroup$
add a comment |
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$begingroup$
I found four references:
- Giné, Evarist, and Joel Zinn. "Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics." Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Birkhäuser Boston, 1992.
- Kondo, Masao, and Hajime Yamato. "Almost sure convergence of a linear combination of U-statistics." Scientiae Mathematicae japonicae 55.3 (2002): 605-613.
- Borovskikh, Yuri Vasilevich. U-statistics in Banach Spaces. VSP, 1996.
- Korolyuk, Vladimir S., and Yuri V. Borovskich. Theory of U-statistics. Vol. 273. Springer Science & Business Media, 2013.
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
$endgroup$
add a comment |
$begingroup$
I found four references:
- Giné, Evarist, and Joel Zinn. "Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics." Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Birkhäuser Boston, 1992.
- Kondo, Masao, and Hajime Yamato. "Almost sure convergence of a linear combination of U-statistics." Scientiae Mathematicae japonicae 55.3 (2002): 605-613.
- Borovskikh, Yuri Vasilevich. U-statistics in Banach Spaces. VSP, 1996.
- Korolyuk, Vladimir S., and Yuri V. Borovskich. Theory of U-statistics. Vol. 273. Springer Science & Business Media, 2013.
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
$endgroup$
add a comment |
$begingroup$
I found four references:
- Giné, Evarist, and Joel Zinn. "Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics." Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Birkhäuser Boston, 1992.
- Kondo, Masao, and Hajime Yamato. "Almost sure convergence of a linear combination of U-statistics." Scientiae Mathematicae japonicae 55.3 (2002): 605-613.
- Borovskikh, Yuri Vasilevich. U-statistics in Banach Spaces. VSP, 1996.
- Korolyuk, Vladimir S., and Yuri V. Borovskich. Theory of U-statistics. Vol. 273. Springer Science & Business Media, 2013.
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
$endgroup$
I found four references:
- Giné, Evarist, and Joel Zinn. "Marcinkiewicz type laws of large numbers and convergence of moments for U-statistics." Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference. Birkhäuser Boston, 1992.
- Kondo, Masao, and Hajime Yamato. "Almost sure convergence of a linear combination of U-statistics." Scientiae Mathematicae japonicae 55.3 (2002): 605-613.
- Borovskikh, Yuri Vasilevich. U-statistics in Banach Spaces. VSP, 1996.
- Korolyuk, Vladimir S., and Yuri V. Borovskich. Theory of U-statistics. Vol. 273. Springer Science & Business Media, 2013.
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
answered Jan 19 '17 at 3:09
MartinMartin
1,1811019
1,1811019
add a comment |
add a comment |
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$begingroup$
They are equivalent. Explanation: $n^kV_n=n(n-1)cdots(n-k+1)U_n+R_n$ where the rest $R_n$ is the sum over every $k$uple with at least two indices equal. There are $O(n^{k-1})$ terms in $R_n$ hence $n^{-k}R_nto0$ almost surely (under various moment hypotheses) hence you are done.
$endgroup$
– Did
Aug 30 '16 at 12:34
$begingroup$
Thanks Did, much appreciated.
$endgroup$
– Martin
Aug 30 '16 at 12:53