On a maximum of stable Lévy processes (Q2752961)

Let \(G_a\) and \(S_a\) be the location and the maximal value, respectively, of an \(\alpha\)-stable Lévy process \(X\) on an interval \([0,a]\). It is shown that the random variables \(S_a(G_a)^{-1/\alpha}\) and \(G_a\) are independent. Using this fact the author studies the location of the maximum of the trajectories \(X(t)\) with low \((S_a<\varepsilon)\) and high \((S_a>1/\varepsilon)\) values of \(S_a\) as \(\varepsilon\to + 0\). It is shown that in the first case the point of maximum is located close to \(0\) whereas in the second case it is located either close to point \(t=a\) if \(\alpha P(X(1)>0)=1, P(X(1)>0)<1\) or is ``spreaded'' over the whole interval if \(\alpha P(X(1)>0)=1\).