10 RICHARD F. BASS AND KRZYSZTOF BURDZY

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). •

Let

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^,

and

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*.