Boundary Crossing of Brownian Motion: Its Relation to the by Hans R. Lerche

By Hans R. Lerche

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

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

This can be a learn document approximately my paintings on sequential statistic~ in the course of 1980 - 1984. subject matters are taken care of that are heavily concerning one another and to the legislations of the iterated logarithm:· I) curved boundary first passage distributions of Brownian movement, eleven) optimum homes of sequential assessments with parabolic and approximately parabolic barriers.

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Then 11. (t) 1jJ (t) ~ cjl( ~ uniformZy on (O,t 1 ] as a + 00. ) (1+0(1)) 47 Proof: We show that for k=O,1,2 (3. 11 ) k e-l/J (t) Je ~(~ )F(dS) uniformlyon (O,t 1 ] as a .... 11) imply that A(t)/B(t) .... 1 uniformly on (O,t 1 ] as a .... 5). 11) follows by a calculation using Laplace's method. ft))O with the constant Q of l/J (t) assumption (11). The c (t) .... and c (t)~ .... oo uniformlyon a a a (O,t 1 ] as a .... 2. For k=O,1,2 holds 00 S-l/J (t) o t JSk~( ~ a satisfy c 'f'a ° ,,1: )f(S)dS 00 l/J (t) l/J (t)k+1 J(1+u)k~(~ u)f(l/Ja(t) (1+u))du a 0 "t We split this integral up into three parts J u<-c a II + f J I u I ::;c a lul $c a J I + 11 + III • u>c a k l/Ja(t) (1+u) ~(~ u)f(l/Ja(t) (1+u))du f(l/Ja(t)) since ca ....

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. 2: Let f be a non-negative and continuous function on D=~X(T,oo) where o i) ii) The following statements are equivalent: O~T

28) ~(~(t)-~(u))du lt-u This equation has an intuitive geometrical meaning for concave boundaries. Let S inf{u>OIW(u)~~(t)-(t-u)~' (t)} denote the first exit time of Brownian motion over the tangent to the curve ~ at t. By concavity holds therefore for all paths with yields ( 3 . 29 ) P (Sed t ) T~t, ~(t)-(t-U)~' holds T~S. 30) ~(~(~t-*(u))dt. 31) P(TEdt) P(SEdt)-P(SEdt,T

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