By Hans R. Lerche

ISBN-10: 0387964339

ISBN-13: 9780387964331

ISBN-10: 1461565693

ISBN-13: 9781461565697

This is a learn record approximately my paintings on sequential statistic~ in the course of 1980 - 1984. subject matters are taken care of that are heavily on the topic of one another and to the legislation of the iterated logarithm:· I) curved boundary first passage distributions of Brownian movement, eleven) optimum houses of sequential exams with parabolic and approximately parabolic limitations. within the first bankruptcy I speak about the tangent approximation for Brownianmotion as an international approximation machine. this is often an extension of Strassen' s method of t'he legislations of the iterated logarithm which connects result of fluctuation conception of Brownian movement with classical tools of sequential facts. within the moment bankruptcy I utilize those connections and derive optimum houses of exams of energy one and repeated value checks for the simpiest version of sequential data, the Brownian movement with unknown float. To either topics:there under1ies an asymptotic strategy that's heavily associated with huge deviation thought: the preventing barriers recede to infinity. this can be a famous process in sequential stötistics that is largely mentioned in Siegmund's fresh ebook ·Sequential Analysis". This procedure additionally results in a few new insights in regards to the legislation of the iterated logarithm (LIL). even if the LIL has been studied for almost seventy years the idea remains to be universal that it applies just for huge sampIe sizes that may by no means be obser ved in practice.

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**Extra resources for Boundary Crossing of Brownian Motion: Its Relation to the Law of the Iterated Logarithm and to Sequential Analysis**

**Example text**

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

2) na(t) is defined as the solution x=na(t) of the implicit equation f(x,t)=a. 10) together yield that the boundaries for t=l satisfy the equations s (2. 14) x ; , t s a Brownian Bridge with endpoints (8,0) and (yo,so) , is mapped into a Brownian motion with drift e starting at (Yo ,~) and running up to So So infinity. 14) and vice versal. The reader is invited to check some of the examples of both sections on their correspondence. These observations raise the further leading question: how are the methods themselves related?

7) below), that the first exit density at the tangent is bigger than that at the curve at time t. In this section we study the asymptotic behaviour of Palt) when a + co for boundaries ~a with strong curvature. 3) Aa (t) 1/1 (t) Palt) = ~ ~(~ ) (1+0(1» holds uniformlyon intervals (O,t 1 ] when a+oo • This statement is studied from different perspectives, at first from an analytic and further below from a probabilistic point of view. 3) has the advantage that it is easy to calculate. Its quality of approximation for finite situations is discussed in the next section.

### 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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