Boundary Crossing of Brownian Motion

Its Relation to the Law of the Iterated Logarithm and to Sequential Analysis

  • Hans Rudolf Lerche

Part of the Lecture Notes in Statistics book series (LNS, volume 40)

Table of contents

  1. Front Matter
    Pages N2-V
  2. Introduction

    1. Hans Rudolf Lerche
      Pages 1-15
  3. Curved Boundary First Passage Distributions of Brownian Motion

    1. Front Matter
      Pages 17-17
    2. Hans Rudolf Lerche
      Pages 33-41
    3. Hans Rudolf Lerche
      Pages 60-76
  4. Optimal Properties of Sequential Tests with Parabolic and Nearly Parabolic Boundaries

  5. Back Matter
    Pages 135-143

About this book


This is a research report about my work on sequential statistic~ during 1980 - 1984. Two themes are treated which are closely related to each other and to the law of the iterated logarithm:· I) curved boundary first passage distributions of Brownian motion, 11) optimal properties of sequential tests with parabolic and nearly parabolic boundaries. In the first chapter I discuss the tangent approximation for Brownianmotion as a global approximation device. This is an extension of Strassen' s approach to t'he law of the iterated logarithm which connects results of fluctuation theory of Brownian motion with classical methods of sequential statistics. In the second chapter I make use of these connections and derive optimal properties of tests of power one and repeated significance tests for the simpiest model of sequential statistics, the Brownian motion with unknown drift. To both topics:there under1ies an asymptotic approach which is closely linked to large deviation theory: the stopping boundaries recede to infinity. This is a well-known approach in sequential stötistics which is extensively discussed in Siegmund's recent book ·Sequential Analysis". This approach also leads to some new insights about the law of the iterated logarithm (LIL). Although the LIL has been studied for nearly seventy years the belief is still common that it applies only for large sampIe sizes which can never be obser­ ved in practice.


Brownian motion Fusion approximation construction derivation distribution equation function functions law of the iterated logarithm likelihood logarithm model statistics themes

Authors and affiliations

  • Hans Rudolf Lerche
    • 1
  1. 1.Institut für Angewandte MathematikHeidelbergFederal Republic of Germany

Bibliographic information

  • DOI
  • Copyright Information Springer-Verlag New York 1986
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-0-387-96433-1
  • Online ISBN 978-1-4615-6569-7
  • Series Print ISSN 0930-0325
  • About this book