8 RICHARD F. BASS AND KRZYSZTOF BURDZY
Suppose Zt is a Bes(3) and Z0 = L\ 0. Let Si = 0. Define
Mi = inf{Zt :t Si},
Ti = i n f { t 5 i :Zt = M1},
Ui =inf{t Ti :Zt = Li).
Since Zt oo a.s. as t —• oo, then C/i oo a.s. We then define inductively
Mi+i = inf{Zt : t Ut},
Ti+i=mi{tUi:Zt=Mi+1},
Li+i=swp{Zt:TitTi+i},
Si+i = inf{i Ui-.Zt = Li+l},
Ui+t = inf{t Ti+i : Zt = Li+1}.
(2.2)
U2 S3 T3
Figure 2.1.
We use Propositions 2.1 and 2.2 to decompose the path of a Bes(3) process.
The Mi will be the future minima of Zt after certain relative maxima, and the Ti
will be the times they occur. The Li are the maxima of the Bes(3) path up until
time Ti and the Si are the times these maxima occur. The Ui are the times Zt
returns to the level Li after hitting a minimum. See Figure 2.1.
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