Small-time behaviour of {L}évy processes
It is proved that, for the Lévy process \(X=\{X_{t},t\geq 0\}\) without the Gaussian component and with spectral measure \(\Pi\) such that \(\Pi(R)=\infty\) and \(\Pi(-\infty,0)>0\), the following statements are equivalent: \[ P(X_{t}>0)\to 1\text{ as }t\downarrow 0;\tag{1} \] \[ X_t/\Delta_t^{(1)}\to \infty\text{ in probability as }t\downarrow0; \tag{2} \] for some deterministic function \(d\) which decreases to \(0\) and is regularly varying of index 1 at 0, \[ X_t/d(t)\to \infty\text{ in probability as }t\downarrow0;\tag{3} \] and \[ A(x)/\sqrt {U(x)M(x)}\to\infty\text{ as }x\downarrow0.\tag{4} \] Here \(M(x)=\Pi(-\infty,-x), x>0\), \(\Delta_{t}^{(1)}\) is the magnitude of the largest negative jump process \((\Delta_s,s>0)\) of \(X\) as \(s\leq t\). \(A(x)\) and \(U(x)\) are certain functions of the tail difference or tail sum of the measure \(\Pi\), respectively.
scientific article; zbMATH DE number 2142135