Page 88 - Special Topic Session (STS) - Volume 3

STS517 Andreas Basse-O’Connor
Since m has finite − ℎ moment we have for all ≥ that
2

and similarly,

First we choose a number q satisfying (1 + ) −1 < < 1, which is possible
since > 0. Next, we choose > {/2, } such that (1 − ) + >
2
2
1, which is possible since < 1. By setting = ( − ) − we obtain the
1
2
following estimate from (2.7), (2.8) and (2.9)

for some > 0 only depending on , and .
2
For the −term from (2.5) we use the following estimate
2

where the second inequality follows by {|| , 1} ≤ || for all ∈ ℝ,

2
which holds since 0 ≤ ≤ . From (2.5), (2.10) and (2.11) we obtain (2.4).
2
In the following we will prove (2.3). By recalling the definition of the
function in (2.6), in the case = , we have by the mean-value theorem
1
1

Hence, if we set = with chosen according to (2.10), we have according
2
2
1
to (2.10) that

which completes the proof of (2.3), and hence the lemma.
Lemma 2.2. Let Z be given by (2.2) and assume that ∈ (1,2) and that the
measure has finite − ℎ moment for some > 0. Then has continuous
sample paths with probability one.
Proof. For each − stable random variable let ‖‖ denote the scale
parameter of . By the isometric property of stable integrals, cf. Proposition
3.4.1 of Samorodnitsky and Taqqu (1994), we have that

77 | I S I W S C 2 0 1 9