By Hans R. Lerche

ISBN-10: 0387964339

ISBN-13: 9780387964331

ISBN-10: 1461565693

ISBN-13: 9781461565697

This is a examine file approximately my paintings on sequential statistic~ in the course of 1980 - 1984. topics are taken care of that are heavily concerning one another and to the legislation of the iterated logarithm:· I) curved boundary first passage distributions of Brownian movement, eleven) optimum homes of sequential checks with parabolic and approximately parabolic barriers. within the first bankruptcy I talk about the tangent approximation for Brownianmotion as a world approximation gadget. this is often an extension of Strassen' s method of t'he legislations of the iterated logarithm which connects result of fluctuation idea of Brownian movement with classical tools of sequential facts. within the moment bankruptcy I utilize those connections and derive optimum homes of exams of strength one and repeated importance checks for the simpiest version of sequential information, the Brownian movement with unknown float. To either topics:there under1ies an asymptotic method that's heavily associated with huge deviation conception: the preventing obstacles recede to infinity. this can be a famous method in sequential stötistics that's widely mentioned in Siegmund's contemporary publication ·Sequential Analysis". This strategy additionally results in a few new insights in regards to the legislation of the iterated logarithm (LIL). even supposing the LIL has been studied for almost seventy years the idea remains to be universal that it applies just for huge sampIe sizes that could by no means be obser ved in practice.

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DDD A similar result holds for the case when F is concentrated on (_00,00). But then the boundaries can be closed. To derive the tangent approximation we put the following conditions on the mixing measure: F has a density fIel with the properties (1) a) there exists a monotone increasing function certain positive constants K and Vo h:~++[',oo) and 46 J (1+Kv)'h(Kv)cjl(v)dv < 00 Vo such that b) The ratio f(6(1tE))/f(6) (lI) for all f(6(1+a))~f(6)h(a) 1 uniformly for all 6 as E + 2-8 ) < sup F(O,y)/(f(y)e Y O

Robbins-Siegmund studied in several papers how the boundaries depend on the underlying mixing measure and constructed boundaries with the most typical growth rates (see p. 36 and 37). The problem, which concave boundaries can be constructed from the two described methods, is to our knowledge not yet solved. 14). 5): denote the measure of the Brownian Bridge process Wo(u) with endpoints (8,0) and (y ,s ). 6). Let Py s 0' (de) 0 SIP (e2°) P (de) >,so Y 1 exp(e-2- -262 /s )p(de) So 0 fp((YeOo's)o)p (de).

14) is also known as the Appelltransformation. Let u satisfy the forward diffusion equation a u=la 2 u t 2 x in a region Du of the space-time plane lR x lR +. Let f(Y,s)=u(~ ,~)/IS~(~). Then f satisfies the backward diffusion equation and vice versa. Thus the Appell-transformation establishes a duality relation between solutions of the forward and backward diffusion equations. 3, the representation theorem of Widder, to one for the harmonie functions of the backward diffusion equation. Here we give aversion due to Robbins-Siegmund ( 1 973) on lR.

### Boundary Crossing of Brownian Motion: Its Relation to the Law of the Iterated Logarithm and to Sequential Analysis by Hans R. Lerche

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