2092 CHAPTER 62. STOCHASTIC PROCESSES

By the Borel Cantelli lemma, there exists a set of measure 0, Nt such that for ω /∈ Nt ,ω iscontained in only finitely many of the sets[

supr∈Q+∩[0,t]

|X (r)| ≥ 2m

]

which shows that for ω /∈ Nt ,supr∈Q+∩[0,t] |X (r)| is bounded. Now let N = ∪∞j=1N j. This

proves the second claim.Next consider the first claim. By the upcrossing estimate, Theorem 60.6.9 or Lemma

60.2.6, and letting a < b and Un[a,b] [c,d] the upcrossings of Yk from a to b on [c,d] for d ≤ t

and c≥ 0,

E(

Un[a,b] [0, t]

)≤ 1

b−aE((Yn−a)+

)=

1b−a

E((X (t)−a)+

).

Hence

P([

Un[a,b] [0, t]≥M

])≤ 1

M

(1

b−aE((X (t)−a)+

)). (62.8.27)

Suppose for some s < t,

lim supr→s+,r∈Q

X (r,ω)> b > a > lim infr→s+,r∈Q

X (r,ω) . (62.8.28)

If this is so, then in (s, t)∩Q there must be infinitely many values of r ∈ Q such thatX (r,ω) ≥ b as well as infinitely many values of r ∈ Q such that X (r,ω) ≤ a. Note thisinvolves the consideration of a limit from one side. Thus, since it is a limit from one sideonly, there are an arbitrarily large number of upcrossings between s and t. Therefore, lettingM be a large positive number, it follows that for all n sufficiently large,

Un[a,b] [0, t] (ω)≥M

which impliesω ∈

[Un[a,b] [0, t]≥M

]which from 62.8.27 is a set of measure no more than

1M

(1

b−aE((X (t)−a)+

)).

This has shown that the set of ω such that for some s ∈ [0, t) 62.8.28 holds is contained inthe set

N[a,b] ≡ ∩∞M=1∪∞

n=1

[Un[a,b] [0, t]≥M

]Now the sets, [

Un[a,b] [0, t]≥M

]