Page 87 - Special Topic Session (STS) - Volume 3

STS517 Andreas Basse-O’Connor
Lemma 2.1, that has cádlág sample paths, and in Lemma 2.2, we will show
that has continuous samples paths. For the basic properties of stable
random variables/processes/measures we refer to Samorodnitsky and Taqqu
(1994). Throughout this section will denote a finite constant which might
vary from line to line.
Lemma 2.1. Suppose that the measure m has finite − ℎ moment for some
> 0. Then the process , given in (2.2), has càdlàg sample paths almost
surely.
Proof. To ease the notation let µ(, ) = () and (, ) =

(−) (0,] (). Moreover, fix > 0. To prove the lemma, we will verify the
existence of > , > /2, > 1/2 and > 1 such that
1
2
2
1

hold for all 0 ≤ ≤ ≤ ≤ , which due to Theorem 4.3 of Basse-
1
2
O’Connor and Rosin´ski (2013) yields that has càdlàg sample paths with
probability one. Let > 0 and ∈ (0, ]. To verify (2.4) we use the mean-
value theorem to obtain

Where

First we will consider the -term from (2.5). Since 0 ≤ ≤ we have that
1
1

which implies that for all > 0,

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