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By Derek F. Lawden

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The second is proved the same way. Since {I(K^ ;7 = 1,2, . . } is an increasing sequence and x is a point in each set K^, X\v[ij^^I(K^ exists as an extended real-valued number and lim^^oo A^j) ^ ^W- /attains its infimum over the closed set Kj. Hence there exists a point x^ in K^ such that I(x^ = I(Kj). } converges to x and by lower semicontinuity *If ATor G equals the empty set (/>, then (c)-(d) hold trivially if we set log QM) = log 0 = ā€” 00 and inf^g^/(x) = 00. 4. Statement of Large Deviation Properties for Levels-1, 2, and 3 37 I(x) < lim I(x,) = lim I{K:) < I(x).

IQ), and that I^'^\P) > 0 with equahty if and only if P = Pp. Thus Ip^\P) measures the discrepancy between P and Pp. It is called the mean relative entropy of P with respect to Pp. Here are some examples of mean relative entropy. 2. P) = alogr ^ Z n,P{(o}'logn,P{co} since n^Pp{co} = r~* for each co e F"^. According to Note 2 of Chapter IX, the Umit A ( P ) = - l i m - Z 7r,P{co}log7r,P{co} exists; /z(P) is called the mean entropy"^ of P. 37) exists and I^^\P) = log r - h(P). By properties of /^^^(P) mentioned above, h(P) < logr and h(P) = logr if and only if P = P^.

Since SJn equals Yj=i ^i^n,h ^J^ is in the set A^ if and only if Lā€ž is in the set of measures B2 = {veM(r):\Y}=iXiVi - rrip] > s}. Hence Qi^^iA^} equals Gli^^ {^2} ā€¢ The level-2 argument just given for the set A2 can be easily modified for the set B2, and we find \imhogQ['^{A,} = l i m i l o g e i ' H ^ 2 } = -min/,^^>(v). We evaluate this minimum in two steps: mm /(2)(v) = min = min min Ih^Kv): ve ^ ( F ) , ^^ x,v^ = z i h^\z). Since / ^ ^(z) = 00 for z ^ [x^, x^], it follows that lim^logaMi}=-min/^^>(z).

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