Central limit theorem when variables are dependent
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I'm wondering if the central limit theorem can be applied when covariance between variables (say time series data) fades quickly enough. For example if there is a sequence of random variables $Z_i$, where $Z_i$ and $Z_j$ are correlated when $|i-j|=1$, can the central limit theorem still be applied to $sqrt{n}frac{1}{n}sum_{i=1}^nZ_i$?
probability statistics
asked Nov 22 at 17:07
James Caribert
192
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up vote
2
down vote
favorite
I'm wondering if the central limit theorem can be applied when covariance between variables (say time series data) fades quickly enough. For example if there is a sequence of random variables $Z_i$, where $Z_i$ and $Z_j$ are correlated when $|i-j|=1$, can the central limit theorem still be applied to $sqrt{n}frac{1}{n}sum_{i=1}^nZ_i$?
probability statistics
asked Nov 22 at 17:07
James Caribert
192
Welcome to MathStackExchange! I would suggest you edit your question to provide some clarifications: What do you mean by CLT? What is the context for this question? Are you taking a course, trying to achieve something or is it a curiosity/fun question?
– Mefitico
Nov 22 at 17:14
1
You may have a look on the chapter 'CLT's for Dependent Variables', especially on 'Mixing Properties', in the book Probability - Theory and Examples of Durrett.
– p4sch
Nov 22 at 17:29
The result you want is probably the CLT for $alpha$-mixing sequences, but it's a bit technical. The short, qualitative answer is that it still holds, but: (1) the growth conditions are a bit stronger; (2) the $Z_i, Z_j$ have to be "approximately independent" as $|i - j| to infty$ (the $alpha$-mixing condition is the precise form of this requirement); (3) in the limiting distribution ${mathcal N}(0, sigma^2)$, the value of $sigma^2$ picks up some extra covariance terms for the $Z_i$.
– anomaly
Nov 22 at 17:42
@p4sch I don't see this in the book. I know there are different versions...maybe it was removed from more recent ones?
– Eric Auld
Nov 22 at 17:47
In the third edition it is chapter 7.7. In fact, in the online version (edition 4.1) this part of the book is not included. (Only 8.8, but not the subsection on mixing properties.)
– p4sch
Nov 22 at 17:56
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I'm wondering if the central limit theorem can be applied when covariance between variables (say time series data) fades quickly enough. For example if there is a sequence of random variables $Z_i$, where $Z_i$ and $Z_j$ are correlated when $|i-j|=1$, can the central limit theorem still be applied to $sqrt{n}frac{1}{n}sum_{i=1}^nZ_i$?
probability statistics
asked Nov 22 at 17:07
James Caribert
192
I'm wondering if the central limit theorem can be applied when covariance between variables (say time series data) fades quickly enough. For example if there is a sequence of random variables $Z_i$, where $Z_i$ and $Z_j$ are correlated when $|i-j|=1$, can the central limit theorem still be applied to $sqrt{n}frac{1}{n}sum_{i=1}^nZ_i$?
probability statistics
probability statistics
asked Nov 22 at 17:07
James Caribert
192
asked Nov 22 at 17:07
James Caribert
192
edited Nov 22 at 17:18
asked Nov 22 at 17:07
James Caribert
192
asked Nov 22 at 17:07
James Caribert
192
asked Nov 22 at 17:07
James Caribert
192
192
Welcome to MathStackExchange! I would suggest you edit your question to provide some clarifications: What do you mean by CLT? What is the context for this question? Are you taking a course, trying to achieve something or is it a curiosity/fun question?
– Mefitico
Nov 22 at 17:14
1
You may have a look on the chapter 'CLT's for Dependent Variables', especially on 'Mixing Properties', in the book Probability - Theory and Examples of Durrett.
– p4sch
Nov 22 at 17:29
The result you want is probably the CLT for $alpha$-mixing sequences, but it's a bit technical. The short, qualitative answer is that it still holds, but: (1) the growth conditions are a bit stronger; (2) the $Z_i, Z_j$ have to be "approximately independent" as $|i - j| to infty$ (the $alpha$-mixing condition is the precise form of this requirement); (3) in the limiting distribution ${mathcal N}(0, sigma^2)$, the value of $sigma^2$ picks up some extra covariance terms for the $Z_i$.
– anomaly
Nov 22 at 17:42
@p4sch I don't see this in the book. I know there are different versions...maybe it was removed from more recent ones?
– Eric Auld
Nov 22 at 17:47
In the third edition it is chapter 7.7. In fact, in the online version (edition 4.1) this part of the book is not included. (Only 8.8, but not the subsection on mixing properties.)
– p4sch
Nov 22 at 17:56
add a comment |
Welcome to MathStackExchange! I would suggest you edit your question to provide some clarifications: What do you mean by CLT? What is the context for this question? Are you taking a course, trying to achieve something or is it a curiosity/fun question?
– Mefitico
Nov 22 at 17:14
1
You may have a look on the chapter 'CLT's for Dependent Variables', especially on 'Mixing Properties', in the book Probability - Theory and Examples of Durrett.
– p4sch
Nov 22 at 17:29
The result you want is probably the CLT for $alpha$-mixing sequences, but it's a bit technical. The short, qualitative answer is that it still holds, but: (1) the growth conditions are a bit stronger; (2) the $Z_i, Z_j$ have to be "approximately independent" as $|i - j| to infty$ (the $alpha$-mixing condition is the precise form of this requirement); (3) in the limiting distribution ${mathcal N}(0, sigma^2)$, the value of $sigma^2$ picks up some extra covariance terms for the $Z_i$.
– anomaly
Nov 22 at 17:42
@p4sch I don't see this in the book. I know there are different versions...maybe it was removed from more recent ones?
– Eric Auld
Nov 22 at 17:47
In the third edition it is chapter 7.7. In fact, in the online version (edition 4.1) this part of the book is not included. (Only 8.8, but not the subsection on mixing properties.)
– p4sch
Nov 22 at 17:56
Welcome to MathStackExchange! I would suggest you edit your question to provide some clarifications: What do you mean by CLT? What is the context for this question? Are you taking a course, trying to achieve something or is it a curiosity/fun question?
– Mefitico
Nov 22 at 17:14
Welcome to MathStackExchange! I would suggest you edit your question to provide some clarifications: What do you mean by CLT? What is the context for this question? Are you taking a course, trying to achieve something or is it a curiosity/fun question?
– Mefitico
Nov 22 at 17:14
1
1
You may have a look on the chapter 'CLT's for Dependent Variables', especially on 'Mixing Properties', in the book Probability - Theory and Examples of Durrett.
– p4sch
Nov 22 at 17:29
You may have a look on the chapter 'CLT's for Dependent Variables', especially on 'Mixing Properties', in the book Probability - Theory and Examples of Durrett.
– p4sch
Nov 22 at 17:29
The result you want is probably the CLT for $alpha$-mixing sequences, but it's a bit technical. The short, qualitative answer is that it still holds, but: (1) the growth conditions are a bit stronger; (2) the $Z_i, Z_j$ have to be "approximately independent" as $|i - j| to infty$ (the $alpha$-mixing condition is the precise form of this requirement); (3) in the limiting distribution ${mathcal N}(0, sigma^2)$, the value of $sigma^2$ picks up some extra covariance terms for the $Z_i$.
– anomaly
Nov 22 at 17:42
The result you want is probably the CLT for $alpha$-mixing sequences, but it's a bit technical. The short, qualitative answer is that it still holds, but: (1) the growth conditions are a bit stronger; (2) the $Z_i, Z_j$ have to be "approximately independent" as $|i - j| to infty$ (the $alpha$-mixing condition is the precise form of this requirement); (3) in the limiting distribution ${mathcal N}(0, sigma^2)$, the value of $sigma^2$ picks up some extra covariance terms for the $Z_i$.
– anomaly
Nov 22 at 17:42
@p4sch I don't see this in the book. I know there are different versions...maybe it was removed from more recent ones?
– Eric Auld
Nov 22 at 17:47
@p4sch I don't see this in the book. I know there are different versions...maybe it was removed from more recent ones?
– Eric Auld
Nov 22 at 17:47
In the third edition it is chapter 7.7. In fact, in the online version (edition 4.1) this part of the book is not included. (Only 8.8, but not the subsection on mixing properties.)
– p4sch
Nov 22 at 17:56
In the third edition it is chapter 7.7. In fact, in the online version (edition 4.1) this part of the book is not included. (Only 8.8, but not the subsection on mixing properties.)
– p4sch
Nov 22 at 17:56
add a comment |
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Welcome to MathStackExchange! I would suggest you edit your question to provide some clarifications: What do you mean by CLT? What is the context for this question? Are you taking a course, trying to achieve something or is it a curiosity/fun question?
– Mefitico
Nov 22 at 17:14
1
You may have a look on the chapter 'CLT's for Dependent Variables', especially on 'Mixing Properties', in the book Probability - Theory and Examples of Durrett.
– p4sch
Nov 22 at 17:29
The result you want is probably the CLT for $alpha$-mixing sequences, but it's a bit technical. The short, qualitative answer is that it still holds, but: (1) the growth conditions are a bit stronger; (2) the $Z_i, Z_j$ have to be "approximately independent" as $|i - j| to infty$ (the $alpha$-mixing condition is the precise form of this requirement); (3) in the limiting distribution ${mathcal N}(0, sigma^2)$, the value of $sigma^2$ picks up some extra covariance terms for the $Z_i$.
– anomaly
Nov 22 at 17:42
@p4sch I don't see this in the book. I know there are different versions...maybe it was removed from more recent ones?
– Eric Auld
Nov 22 at 17:47
In the third edition it is chapter 7.7. In fact, in the online version (edition 4.1) this part of the book is not included. (Only 8.8, but not the subsection on mixing properties.)
– p4sch
Nov 22 at 17:56