By AAD W VAN DER VAART
This publication attempts to do 3 issues. the 1st aim is to provide an exposition of yes modes of stochastic convergence, particularly convergence in distribution. The classical idea of this topic used to be constructed regularly within the Nineteen Fifties and is easily summarized in Billingsley (1968). over the last 15 years, the necessity for a extra common conception permitting random parts that aren't Borel measurable has develop into good validated, quite in constructing the idea of empirical tactics. half 1 of the e-book, Stochastic Convergence, offers an exposition of this sort of conception following the guidelines of J. Hoffmann-J!1Jrgensen and R. M. Dudley. A moment aim is to take advantage of the vulnerable convergence thought heritage devel oped partly 1 to give an account of significant elements of the trendy idea of empirical procedures listed via periods of units and features. The susceptible convergence idea constructed partially 1 is critical for this, just because the empirical techniques studied partially 2, Empirical strategies, are obviously seen as taking values in nonseparable Banach areas, even within the most simple situations, and are usually now not Borel measurable. a lot of the speculation offered partially 2 has formerly been scattered within the magazine literature and has, hence, been available in basic terms to a comparatively small variety of experts. In view of the significance of this concept for statis tics, we are hoping that the presentation given right here will make this thought extra available to statisticians in addition to to probabilists attracted to statistical applications.
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Additional resources for Weak Convergence and Empirical Processes: With Applications to Statistics
For asymptotically tight nets, the situation is nice - asymptotic tightness plus only a little bit of measurability gives asymptotic measurability. The next lemma is an abstract version of this result. Interesting special cases are discussed in the next sections. 13 Lemma. Let the net Xo. t Then the net Xo. is asymptotically measurable. Proof. Fix c > 0 and a compact K such that limsupP*(Xo. ~ Kti) ::; c for every 8 > O. Assume without loss of generality that :F contains the constant functions.
Then the net (X"" Y",) is asymptotically tight and asymptotically measurable. The limit points have marginals X and c and so are uniquely determined as (X, c). ) If X", and Y", take their values in a fixed topological vector space, the previous result can be combined with the continuous mapping theorem to obtain Slutsky's theorem: if X", "'" X and Y", "'" c with X separable and c a constant, then X", + Y", "'" X + c. Similarly, under the same conditions, X",Y", "'" cX in the case Y", are scalars; and provided c > 0, X",/Y", "'" X/c also.
Then E· f(X",) = E· f(X",) for every bounded f: II} f-+ K Consequently, whenever one has to do with maps X", and is interested only in their "laws," it is no loss of generality to assume that the maps are defined on a single probability space. 5. Let d and e be metrics on the same set II} with the following property: if Xn -+ x for e and a limit x E lI}o, then Xn -+ x for d; for every sequence Xn and a given subset lI}o of j[)). ".. X for e, where X takes its values in lI}o, implies the same for d.
Weak Convergence and Empirical Processes: With Applications to Statistics by AAD W VAN DER VAART