How can I prove the asymptotic equipartition property (AEP) for an identically distributed markov chain?
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$ $ Hi, everyone. I am recently reading the lecture note of EE376a : Information Theory course from Stanford University. This note introduces that we can prove the Asymptotic Equipartition Property (or the Shannon-McMillan-Breiman Theorem) for the case that the given stochastic process $left{ X_n right}$ is a time-invariant discrete-time Markov chain with a finite state space $mathcal{X}$ such that every $X_n$ is identically distributed over $mathcal{X}$ with the distributuion $p$. The following is the statement. :
$$-frac{1}{n} log p(X_1, cdots, X_n) rightarrow H(X_2|X_1)$$
in probability as $n rightarrow infty$. The note introduces the proof of this statement by using Weak Law of Large Numbers for weak dependency. :
Let ${ Y_n }$ be a sequence of identically distributed random variables such that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0.$$ Then, $frac{1}{n} sum_{i=1}^{n} Y_i rightarrow mathbb{E}[Y_1]$ in probability as $n rightarrow infty$.
Thus, we let $Y_k := log p(X_k |X_{k-1})$ for $k geq 2$ and $Y_1 := log p(X_1)$. Therefore, it suffices to verify that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0 cdots (star)$$
and the statement comes from the WLLN for weak dependency. Now, here is my question. How can I prove the part $(star)$? Please, share some good ideas for this problem.
probability statistics information-theory
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
$endgroup$
add a comment |
$begingroup$
$ $ Hi, everyone. I am recently reading the lecture note of EE376a : Information Theory course from Stanford University. This note introduces that we can prove the Asymptotic Equipartition Property (or the Shannon-McMillan-Breiman Theorem) for the case that the given stochastic process $left{ X_n right}$ is a time-invariant discrete-time Markov chain with a finite state space $mathcal{X}$ such that every $X_n$ is identically distributed over $mathcal{X}$ with the distributuion $p$. The following is the statement. :
$$-frac{1}{n} log p(X_1, cdots, X_n) rightarrow H(X_2|X_1)$$
in probability as $n rightarrow infty$. The note introduces the proof of this statement by using Weak Law of Large Numbers for weak dependency. :
Let ${ Y_n }$ be a sequence of identically distributed random variables such that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0.$$ Then, $frac{1}{n} sum_{i=1}^{n} Y_i rightarrow mathbb{E}[Y_1]$ in probability as $n rightarrow infty$.
Thus, we let $Y_k := log p(X_k |X_{k-1})$ for $k geq 2$ and $Y_1 := log p(X_1)$. Therefore, it suffices to verify that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0 cdots (star)$$
and the statement comes from the WLLN for weak dependency. Now, here is my question. How can I prove the part $(star)$? Please, share some good ideas for this problem.
probability statistics information-theory
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
$endgroup$
add a comment |
$begingroup$
$ $ Hi, everyone. I am recently reading the lecture note of EE376a : Information Theory course from Stanford University. This note introduces that we can prove the Asymptotic Equipartition Property (or the Shannon-McMillan-Breiman Theorem) for the case that the given stochastic process $left{ X_n right}$ is a time-invariant discrete-time Markov chain with a finite state space $mathcal{X}$ such that every $X_n$ is identically distributed over $mathcal{X}$ with the distributuion $p$. The following is the statement. :
$$-frac{1}{n} log p(X_1, cdots, X_n) rightarrow H(X_2|X_1)$$
in probability as $n rightarrow infty$. The note introduces the proof of this statement by using Weak Law of Large Numbers for weak dependency. :
Let ${ Y_n }$ be a sequence of identically distributed random variables such that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0.$$ Then, $frac{1}{n} sum_{i=1}^{n} Y_i rightarrow mathbb{E}[Y_1]$ in probability as $n rightarrow infty$.
Thus, we let $Y_k := log p(X_k |X_{k-1})$ for $k geq 2$ and $Y_1 := log p(X_1)$. Therefore, it suffices to verify that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0 cdots (star)$$
and the statement comes from the WLLN for weak dependency. Now, here is my question. How can I prove the part $(star)$? Please, share some good ideas for this problem.
probability statistics information-theory
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
$endgroup$
$ $ Hi, everyone. I am recently reading the lecture note of EE376a : Information Theory course from Stanford University. This note introduces that we can prove the Asymptotic Equipartition Property (or the Shannon-McMillan-Breiman Theorem) for the case that the given stochastic process $left{ X_n right}$ is a time-invariant discrete-time Markov chain with a finite state space $mathcal{X}$ such that every $X_n$ is identically distributed over $mathcal{X}$ with the distributuion $p$. The following is the statement. :
$$-frac{1}{n} log p(X_1, cdots, X_n) rightarrow H(X_2|X_1)$$
in probability as $n rightarrow infty$. The note introduces the proof of this statement by using Weak Law of Large Numbers for weak dependency. :
Let ${ Y_n }$ be a sequence of identically distributed random variables such that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0.$$ Then, $frac{1}{n} sum_{i=1}^{n} Y_i rightarrow mathbb{E}[Y_1]$ in probability as $n rightarrow infty$.
Thus, we let $Y_k := log p(X_k |X_{k-1})$ for $k geq 2$ and $Y_1 := log p(X_1)$. Therefore, it suffices to verify that
$$lim_{n rightarrow infty} frac{1}{n^2} sum_{i=1}^{n} sum_{j=1}^{n} Cov[Y_i, Y_j] = 0 cdots (star)$$
and the statement comes from the WLLN for weak dependency. Now, here is my question. How can I prove the part $(star)$? Please, share some good ideas for this problem.
probability statistics information-theory
probability statistics information-theory
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
edited Jan 3 at 13:52
shannonentropy
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
asked Jan 2 at 10:59
shannonentropyshannonentropy
62
62
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