Asymptotic equipartition property

Summary

In information theory, the asymptotic equipartition property (AEP) is a general property of the output samples of a stochastic source. It is fundamental to the concept of typical set used in theories of data compression. Roughly speaking, the theorem states that although there are 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. One way of intuitively understanding the property is through Cramér's large deviation theorem, which states that the probability of a large deviation from mean decays exponentially with the number of samples. Such res

Official source

https://en.wikipedia.org/wiki/Asymptotic_equipartition_property

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We study the optimal strategy for a sailboat to reach an upwind island under the hypothesis that the wind direction fluctuates according to a Brownian motion and the wind speed is constant. The work is motivated by a concrete problem which typically arises during sailing regattas, namely finding the best tacking strategy to reach the upwind buoy as quickly as possible. We assume that there is no loss of time when tacking. We first guess an optimal strategy and then we establish its optimality by using the dynamic programming principle. The Hamilton Jacobi Bellmann equation obtained is a parabolic PDE with Neumann boundary conditions. Since it does not admit a closed form solution, the proof of optimality involves an intricate estimate of derivatives of the value function. We explicitly provide the asymptotic shape of the value function. In order to do so, we prove a result on large time behavior for solutions to time dependent parabolic PDE using a coupling argument. In particular, a boat far from the island approaches the island at

\frac{1}{2} + \frac{\sqrt{2}}{\pi} = 95.02\%

of the boat's speed.

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\frac{1}{2} + \frac{\sqrt{2}}{\pi} = 95.02\%

of the boat's speed.

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