CUTTING BROWNIAN PATHS 11

Since d is an exponential with parameter 1, then e~Ci is uniform on [0,1], hence so

is 1 — e~Ci. This implies Bi is an exponential with parameter 1. Taking logarithms

of (2.5) proves the proposition. •

A distribution is called bilateral exponential if it has a density given by

e-\x\ j2 for X G R . By Feller (1971), p. 49, Ai — Bi is a bilateral exponential, and

so we have

Theorem 2.5. Vm — V\ is the sum of i.i.d. bilateral exponentials.