AEP For Non-stationary Discrete-time Source Producing Independent Symbols
The assumptions of stationarity/ergodicity/identical 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 for all i, for some M>0, the following holds (AEP):
where
Read more about this topic: Asymptotic Equipartition Property
Famous quotes containing the words source, producing, independent and/or symbols:
“To not know would be a source of pain.”
—Sophocles (497–406/5 B.C.)
“In an industrial society which confuses work and productivity, the necessity of producing has always been an enemy of the desire to create.”
—Raoul Vaneigem (b. 1934)
“The love of the famous, like all strong passions, is quite abstract. Its intensity can be measured mathematically, and it is independent of persons.”
—Susan Sontag (b. 1933)
“As usual I finish the day before the sea, sumptuous this evening beneath the moon, which writes Arab symbols with phosphorescent streaks on the slow swells. There is no end to the sky and the waters. How well they accompany sadness!”
—Albert Camus (1913–1960)