An Introduction to the Theory of Large Deviations by D.W. Stroock

By D.W. Stroock

These notes are in response to a path which I gave throughout the educational 12 months 1983-84 on the collage of Colorado. My goal was once to supply either my viewers in addition to myself with an creation to the idea of 1arie deviations • The association of sections 1) via three) owes anything to likelihood and very much to the superb set of notes written by means of R. Azencott for the direction which he gave in 1978 at Saint-Flour (cf. Springer Lecture Notes in arithmetic 774). To be extra unique: it's likelihood that i used to be round N. Y. U. on the time'when M. Schilder wrote his thesis. and so it can be thought of likelihood that I selected to exploit his outcome as a leaping off aspect; with merely minor adaptations. every little thing else in those sections is taken from Azencott. specifically. part three) is little greater than a rewrite of his exoposition of the Cramer conception through the information of Bahadur and Zabel. in addition. the short therapy which i've got given to the Ventsel-Freidlin idea in part four) is back in line with Azencott's rules. All in all. the largest distinction among his and my exposition of those themes is the language during which we've got written. besides the fact that. one other significant distinction has to be pointed out: his bibliography is large and constitutes an outstanding advent to the on hand literature. mine stocks neither of those attributes. beginning with part 5).

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E. e. not just compact ones). n n - xEF' ~ The next result provides us with our first step toward removing these Tim l log n~ deficiencies. 26) Theorem: L >0 Let E I). c: there is a compact for each L all closed Proof: >0 F , and {x : A- (x) ~ E -< L} lim l log ~ (F) n n n~ be as above. ~ such that {x : A- ex) ~ -< L} TIiii l log ~ (~) < -L n n is a compact convex set. < -inf xEF Since we already know that have proved that Assume that for each n~ Then Moreover, for A- (F) ~ A- ~ is 1. s. c. and convex, we will is compact and convex as soon as we show that it is contained in a compact set.

L 6 ) j. ~) ~ Iln(l. L {, j. ~} n CL). Thus n n I xm n I xm is continuous on therefore: l). cc E r(f;~), TIiii .!. log Il (Ie) < -L n~ n n -1. 26) The rest of the theorem now follows from Theorems o Having devoted so much effort to derive a large deviations result associated with the law of large numbers for Banach space valued random variables. it seems only right to see that we have in fact proved the strong law of large numbers for such random variables. 35) Theorem (Ranga Rao): E be a separable Banach space.

The second point is a little more interesting. 16) to generalize the argument used at the end of the proof of Theorem The only difficulty in doing so is overcome by the following lemma. 25) Lemma: Let E and is a compact set contained in compact. In particular, if H and if convex A H v Then, the closed convex hull such that v(A) - v(K) v(A v(A) - v(K) >0 given 0 K is supported on there is a compact Suppose that we have proved the first part. n H) K

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