The asymptotic equipartition property (AEP) is a general property used extensively in information theory concerning the output samples of a stochastic source. It is fundamental to the concept of typical set used in theories of compression.

Roughly speaking, the theorem states that although there are very many series of results that may be produced by a random process, the one actually produced is most probably from a loosely-defined set of outcomes that all have approximately the same chance of being the one actually realized. (This is a consequence of the law of large numbers and ergodic theory.) Although there are individual outcomes which have a higher probability than any outcome in this set, the vast number of outcomes in the set almost guarantees that the outcome will come from the set.

In the field of Pseudorandom number generation, a candidate generator of undetermined quality whose output sequence lies too far outside the typical set by some statistical criteria is rejected as insufficiently random. Thus, although the typical set is loosely defined, pratical notions arise concerning sufficient typicality.

Definition

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Given a discrete-time stationary ergodic stochastic process $X$ on the probability space $(\backslash Omega,B,p)$, AEP is an assertion that

$-\backslash frac\{1\}\{n\}\; \backslash log\; p(X\_1^n)\; \backslash to\; H(X)$

where $X\_1^n$ denotes the process limited to duration $\backslash \{1,\backslash dots,n\backslash \}$, and $H(X)$ or simply $H$ denotes the entropy rate of $X$, which must exist for all discrete-time stationary processes including the ergodic ones. AEP is proved for finite-valued (i.e. ) stationary ergodic stochastic processes in the Shannon-McMillan-Breiman theorem using the ergodic theory and for any i.i.d. sources directly using the law of large numbers in both the discrete-valued case (where $H$ is simply the entropy of a symbol) and the continuous-valued case (where $H$ is the differential entropy instead). The definition of AEP can also be extended for certain classes of continuous-time stochastic processes for which a typical set exists for long enough observation time. The convergence is proven almost sure in all cases.

AEP for discrete-time i.i.d. sources

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Given $X$ is an i.i.d. source, its time series X₁, ..., X_n is i.i.d. with entropy H(X) in the discrete-valued case and differential entropy in the continuous-valued case. The weak law of large numbers gives the AEP with convergence in probability,

$$

\lim_{n\to\infty}\Pr\left\log p(X_1, X_2, ..., X_n) - H(X)\right|< \epsilon\right=0 \qquad \forall \epsilon>0. since the entropy is equal to the expectation of $\backslash frac\{1\}\{n\}\; \backslash log\; p(X\_1,\; X\_2,\; ...,\; X\_n)$.

The strong law of large number asserts the stronger almost sure convergence,

$$

\Pr\left- \frac{1}{n} \log p(X_1, X_2, ..., X_n) = H(X)\right=1 which implies the result from the weak law of large numbers.

AEP for discrete-time finite-valued stationary ergodic sources

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Consider a finite-valued sample space $\backslash Omega$, i.e. , for the discrete-time stationary ergodic process $X:=\backslash \{X\_n\backslash \}$ defined on the probability space $(\backslash Omega,B,p)$. The AEP for such stochastic source, known as the Shannon-McMillan-Breiman theorem, can be shown using the sandwich proof by Algoet and Cover outlined as follows:

• Let $x$ denote some measurable set $x=X(A)$ for some $A\backslash in\; B$
• Parameterize the joint probability by $n$ and x as $j(n,x):=p(x\_0^\{n-1\})$
• Parameterize the conditional probability by $i,k$ and $x$ as $c(i,k,x)\; :=\; p(x\_i|x\_\{i-k\}^\{i-1\})$.
• Take the limit of the conditional probability as $k\backslash to\backslash infty$ and denote it as $c(i,x):=p(x\_i|x\_\{-\backslash infty\}^\{i-1\})$
• Argue the two notions of entropy rate $\backslash lim\_\{n\backslash to\backslash infty\}\; Ej(n,X)$ and $\backslash lim\_\{n\backslash to\backslash infty\}\; Ec(n,n,X)$ exist and are equal for any stationary process including the stationary ergodic process $X$. Denote it as $H$.
• Argue that both $c(i,k,X):=\backslash \{p(X\_i|X\_\{i-k\}^\{i-1\})\backslash \}$ and $c(i,X):=\backslash \{p(X\_i|X\_\{-\backslash infty\}^\{i-1\})\backslash \}$, where $i$ is the time index, are stationary ergodic processes, whose sample means converge almost surely to some values denoted by $H^k$ and $H^\{\backslash infty\}$ respectively.
• Define the $k$-th order Markov approximation to the probability $a(n,k,x)$ as

$a(n,k,x):=p(X\_0^\{k-1\})\backslash prod\_\{i=k\}^\{n-1\}p(X\_i|X\_\{i-k\}^\{i-1\})=j(k,x)\backslash prod\_\{i=k\}^\{n-1\}\; c(i,k,x)$

• Argue that $a(n,k,X(\backslash Omega))$ is finite (although it may not be 1 and so Markov approximation may not be a valid probability measure) from the finite-value assumption.
• Express $-\backslash frac1n\backslash log\; a(n,k,X)$ in terms of the sample mean of $c(i,k,X)$ and show that it converges almost surely to $H^k$
• Define $a(n,x):=p(x\_0^\{n-1\}|x\_\{-\backslash infty\}^\{-1\})$, which is a probability measure.
• Express $-\backslash frac1n\backslash log\; a(n,X)$ in terms of the sample mean of $c(i,X)$ and show that it converges almost surely to $H^\{\backslash infty\}$
• Argue that $H^k\backslash searrow\; H$ as $k\backslash to\backslash infty$ using the stationarity of the process.
• Argue that $H=H^\{\backslash infty\}$ using the Lévy's martingale convergence theorem and the finite-value assumption.
• Show that $E\backslash left*=a(n,\; k,X(\backslash Omega))$ which is finite as argued before.
• Show that $E\backslash left*=1$ by conditioning on the infinite past $X\_\{-\backslash infty\}^\{-1\}$ and iterating the expectation.
• Show that $(\backslash forall\; \backslash alpha\backslash in\backslash mathbb\{R\})\backslash left(\backslash Pr\backslash left\backslash alpha\; \backslash right\backslash leq\; \backslash frac\{a(n,\; k,X(\backslash Omega))\}\{\backslash alpha\}\backslash right)$ using the Markov's inequality and the expectation derived previously.
• Similarly, show that $(\backslash forall\; \backslash alpha\backslash in\backslash mathbb\{R\})\backslash left(\backslash Pr\backslash left\backslash alpha\; \backslash right\backslash leq\; \backslash frac\{1\}\{\backslash alpha\}\backslash right)$, which is equivalent to $(\backslash forall\; \backslash alpha\backslash in\backslash mathbb\{R\})\backslash left(\backslash Pr\backslash left\backslash frac1n\backslash log\backslash alpha\; \backslash right\backslash leq\; \backslash frac\{1\}\{\backslash alpha\}\backslash right)$.
• Show that $\backslash limsup\_\{n\backslash to\backslash infty\}$ of both $\backslash frac1n\; \backslash log\; \backslash frac\{a(n,k,X)\}\{j(n,X)\}$ and $\backslash frac1n\; \backslash log\backslash frac\{j(n,X)\}\{a(n,X)\}$ are non-positive almost surely by setting $\backslash alpha=n^\{\backslash beta\}$ for any $\backslash beta>1$ and applying the Borel-Cantelli lemma.
• Show that $\backslash liminf$ and $\backslash limsup$ of $-\backslash frac1n\; \backslash log\; j(n,X)$ are lower and upper bounded almost surely by $H^\{\backslash infty\}$ and $H^k$ respectively by breaking up the logarithms in the previous result.
• Complete the proof by pointing out that the upper and lower bounds are shown previously to approach $H$ as $k\backslash to\backslash infty$.

AEP for non-stationary discrete-time source producing independent symbols

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The assumptions of stationarity/ergodicity/idential distribution of random variables is not essential for the AEP to hold. Indeed, as is quite clear intuitively, the AEP requires only some form of the law of large numbers to hold, which is fairly general. However, the expression needs to be suitably generalized, and the conditions need to be formulated precisely.

We assume that the source is producing independent symbols, with possibly different output statistics at each instant. We assume that the statistics of the process are known completely, that is, the marginal distribution of the process seen at each time instant is known. The joint distribution is just the product of marginals. Then, under the condition (which can be relaxed) that

where, $\backslash bar\{H\_n\}(X)=\backslash frac\{H(X\_1,X\_2,\backslash ldots,X\_n)\}\{n\}$

• Proof

The proof follows from a simple application of Markov's inequality (applied to second moment of $log(p(X\_i))$.

It is obvious that the proof holds if any moment is uniformly bounded for $r>1$ (again by Markov's inequality applied to rth moment). $\backslash Box\{\}$

Even this condition is not necessary, but given a non-stationary random process, it should not be difficult to test whether the AEP holds using the above method.

Applications for AEP for non-stationary source producing independent symbols

The AEP for non-stationary discrete-time independent process leads us to (among other results) source coding theorem for non-stationary source (with independent output symbols) and channel coding theorem for non-stationary memoryless channels.

Source Coding Theorem

The source coding theorem for discrete time non-stationary independent sources can be found here: source coding

Channel Coding Theorem

Channel coding theorem for discrete time non-stationary memoryless channels can be found here: noisy channel coding theorem

AEP for certain continuous-time stationary ergodic sources

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Discrete-time functions can be interpolated to continuous-time functions. If such interpolation $f$ is measurable, we may define the continuous-time stationary process accordingly as $\backslash tilde\{X\}:=f\backslash circ\; X$. If AEP holds for the discrete-time process, as in the i.i.d. or finite-valued stationary ergodic cases shown above, it automatically holds for the continuous-time stationary process derived from it by some measurable interpolation. i.e. $-\backslash frac\{1\}\{n\}\; \backslash log\; p(\backslash tilde\{X\}\_0^\backslash tau)\; \backslash to\; H(X)$ where $n$ corresponds to the degree of freedom in time $\backslash tau$. $nH(X)/\backslash tau$ and $H(X)$ are the entropy per unit time and per degree of freedom respectively, defined by Shannon.

An important class of such continuous-time stationary process is the bandlimited stationary ergodic process with the sample space being a subset of the continuous $\backslash mathcal\{L\}\_2$ functions. AEP holds if the process is white, in which case the time samples are i.i.d., or there exists $T>1/2W$, where $W$ is the nominal bandwidth, such that the $T$-spaced time samples take values in a finite set, in which case we have the discrete-time finite-valued stationary ergodic process.

Any time-invariant operations also preserves AEP, stationarity and ergodicity and we may easily turn a stationary process to non-stationary without losing AEP by nulling out a finite number of time samples in the process.

See also

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• Source coding theorem

References

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The Classic Paper

• Claude E. Shannon. A Mathematical Theory of Communication. Bell System Technical Journal, July/October 1948.

Other Journal Articles

• Paul H. Algoet and Thomas M. Cover. A Sandwich Proof of the Shannon-McMillan-Breiman Theorem. The Annals of Probability, 16(2): 899-909, 1988.

Textbooks on Information Theory

• Thomas M. Cover, Joy A. Thomas. Elements of information theory New York: Wiley, 1991. ISBN 0471062596

Information theory