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:
“Man is a stream whose source is hidden. Our being is descending into us from we know not whence. The most exact calculator has no prescience that somewhat incalculable may not balk the very next moment. I am constrained every moment to acknowledge a higher origin for events than the will I call mine.”
—Ralph Waldo Emerson (1803–1882)
“Syntax is the study of the principles and processes by which sentences are constructed in particular languages. Syntactic investigation of a given language has as its goal the construction of a grammar that can be viewed as a device of some sort for producing the sentences of the language under analysis.”
—Noam Chomsky (b. 1928)
“Labor is prior to, and independent of, capital. Capital is only the fruit of labor, and could never have existed if labor had not first existed. Labor is the superior of capital, and deserves much the higher consideration.”
—Abraham Lincoln (1809–1865)
“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)