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Let the set V = {U: U is a x-open neighborhood of Q}be directed by U>V iff U c v. With this (partial) ordering on the set V, {f^: U e V} and (Qy: U € P} are nets in F and A respectively. Since compact-open topology, the net { f ^ Let for x = (x^^ llxll = maxi

1. From the preceding proof it is clear that the condition "F assumes infinitely many values" is redundant in Sanov's theorem. The crucial property of the e-neighborhoods Vm seems to be that K(V ,F) = Kn (V ,F) for the partition P corresponding to V . 1 that we do not have to impose this condition on the sets V m and that we can look at the quantities Kp(vm 'F ) directly for suitably chosen sets V m which do not necessarily have the property K„(V ,F) = K(V ,F). 5. LINEAR FUNCTIONS OF EMPIRICAL PMS Several important statistics are in fact linear functions of empirical pms.

If ft c ft c d , then f t ^ = {G € ft: nG(x) e 7L for all x e JR }. A set will be called x-open if the set of pms {P e A . : G e ft} is open in G 1 the topology x defined on A ^ . The topology x on D is defined by these x-open d sets. ) , B = Ex, ,x B = [x ,x ), B = [x ,“) , where l i z 1 z m-l m- 2 m- 1 m m- 1 -oo < x. < x- <. < x „ < °°. Let the set W be defined by 1 2 m-l m 1 W = U1? (B. x [a. ,b. ]) , where 0 = a. < a 0 < ... < a < 1 , 0 < b „ < b 0 <... ,b - a > 0, b . - a > 0. , if x e B , .