By Te Sun Han

This booklet opens a singular size within the 50 yr background of mathematical theories of "information" because the beginning of Shannon thought. to begin with, it introduces, rather than the conventional proposal of entropy and mutual info, the thoroughly new and hugely unconventional process of "information-spectrum" as a simple yet robust device for developing the final thought of data. Reconstructiong step by step all of the crucial significant subject matters in info conception from the point of view of such an "information-spectrum", this entire paintings offers an obtainable advent to the recent form of mathematical thought of knowledge that focuses typically on normal nonstationary and /or nonergodic assets and channels, in transparent distinction with the conventional theories of knowledge. This ebook is a brand new non-traditional theoretical reference for communique execs and statisticians focusing on info idea.

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These coiisideratioiis lead us to introduce the precise coiicept of mutual iiidependeiice of a collection of events by not pairwise but by systems of equations so that the above anomaly cannot occur. Definition 1 Let ( R , C, P) be a probability space and {Ai, i E I ) c P(0) be a family of events. They are said to be pairwise independent if for each distinct i, j in I we have P(Ai n Aj) = P(A,)P(Aj). If A,,, . . ,n. , equations (1) hold for each n 2. ,m , 1 < m 5 n. Thus if Ai E Ai then {Ai, i E I ) is a mutually independent family.

4. > Definition 1 Let {X, X n , n 1) be a family of random variables on a probability space (R, C, P ) . e. if there is a set N E C, P ( N ) = 0 aiid Xn(w) + X ( w ) , as n + oo, for eachw E R - N . (b) The sequence is said t o converge t o X in probability if for each E > 0, > P we have limn,, P [ X n - XI E ] = 0, symbolically written as Xn + X (or as p limn Xn = X ) . (c) The sequence is said t o converge in distribution t o X , often written D Xn + X if FX,, (z) + &(z) at all points z E R for which z is a continuity poiiit of Fx, where FX,, , FX are distribution functions of X, and X (cf.

Some of these have already appeared in Chapter 1. We state them again and give some interrelations here. It turns out that for sums of independent (integrable) random variables, these are all equivalent, but this is a relatively deep result. A partial solution is given in Problem 16. Several inequalities are needed for the proof of the general case. We start with the basic Kolmogorov inequality aiid a few of its variants. As consequences, some important "strong limit laws" will be established. 4.