An almost sure functional limit theorem at zero for a class of Lévy processes normed by the square root function, and applications
Let \(X\) be a Levy process with characteristic triple \((\gamma, \sigma^2=0, \Pi)\) and assume that \(V(x)=\int_{| y| \leq x} y^2 \Pi(dy) > 0\) for all \(x>0\). In a recent result of Bertoin, Doney and Maller it is stated that if \[ \lambda^*_X := \inf \{ a>0 : \int_0^1 \exp(-\frac{a^2}{2V(x)}) \frac{dx}{x} < \infty \} \in [0,\infty] , \] then almost sure \[ -\liminf_{t\downarrow 0} \frac{X(t)}{\sqrt t} = \limsup_{t\downarrow 0} \frac{X(t)}{\sqrt t} = \lambda^*_X . \] This in some way surprising result is supposed to hold in the present paper: let \(\lambda^*_X \in (0,\infty)\). Then the authors give a functional version (Theorem 2) of the statement above. Using a kind of continuous mapping theorem (Theorem 3), they characterize the almost sure cluster sets for a variety of random processes under which: (i) the interpolated process \(Z^X_t(y) := \frac{X(ty)}{\lambda^*_X \sqrt t} , \;0\leq y\leq 1 \); (ii) the reflected processes \(R(t):= (\sup_{s\leq t}X(s)) - X(t) , \;r(t):= X(t) - (\inf_{s\leq t}X(s)) \); (iii) the process \(L\) satisfying \(L(u)= \int_0^u B(L(t))\,dt + X(u) , \;0\leq u \leq 1\), where \(B\) is a Lipschitz function. Rigorous proofs of all statements are given in the last three sections.
scientific article; zbMATH DE number 5351827
scientific article