last remark with V = Li+\ M*+i, A' = A M*+i, and £ = Li M$+i, to conclude
that the conditional distribution of log[(Lj+i Mi+i)/(Li Mi+i)] is exponential
with parameter 1. Since the conditional distribution does not depend on the values
of Li and Mi+i, we obtain (e).
Vi = \og(Li - Mi). (2.3)
The next two theorems say that Vi is the sum of i.i.d. bilateral exponentials.
Theorem 2.4. Let Ai, Bi be two independent sequences of independent identically
distributed exponential random variables with parameter 1. Then {Vm Vi,m 1}
has the same law as {]C2a {A% Bi), m 1}-
Proof. Let
By Proposition 2.3, Ci and Ai are exponentials with parameter 1. We next
show independence. By Proposition 2.1(c) and the pseudo-strong Markov property
at Ui, the process Zt+Ui Mi is a Bes(3) started at Li Mi. Its law depends
on L i , . . . , Lj, M i , . . . , Mi only through the value of Li Mi. By scaling, Zt =
{Zt+Ui —Mi) I {Li Mi) has the law of a Bes(3) started at 1, and is thus independent
of I q , . . . , Li, M i , . . . , Mi. Since e~Ci is the minimum of Zt, then d is independent
of A i , . . . , j4i_i,Ci,... , Ci_i.
To see that A; is independent of A\,..., Ai-\, C\,..., Ci, we use the pseudo-
strong Markov property at time Vi. By Proposition 2.1(b), the law of Zt+Ui Mi,
t Ti+i Ui, conditional on having minimum M*+i Mi at time T
i +
i Ui, is
that of a Brownian motion started at L; M* and killed on hitting M^+i M$.
By scaling, Zt = (^t+c/i M;+i)/(Li M;+i) has the law of a Brownian motion
started at 1 and killed on hitting 0. Since eAi is the maximum of Zt, then Ai is
independent of L\,..., Li, M\,..., M^+i, and hence of A\,..., Ai-\, C\,..., C*.
We have
Mi+i = ( ^ - M;)e- C i + M^,
Li+i = ( ^ - M
i +
i)e A i + M
i +
i .
Let Bi = - log(l - e~ C i ). We then have
ev^ = Ll+l - Mw = (Li - Ml+l)eA* (2.5)
- (Li - [(Li - Mt)e-C* + Mi])eAi
= (Li - Mi)(l - e~c*)eA*
= ev*e-B*eA*.
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