Compactness and continuity properties for a Lévy process at a two-sided exit time. (English) Zbl 1440.62066

Maller, Ross A.; Mason, David M.

Electron. J. Probab. 25, Paper No. 51, 26 p. (2020).

Summary: We consider a Lévy process \(X=(X(t))_{t\geq 0}\) in a generalised Feller class at 0, and study the exit position, \(\left\vert X(T(r))\right\vert\), as \(X\) leaves, and the position, \(\left\vert X(T(r)-)\right\vert\), just prior to its leaving, at time \(T(r)\), a two-sided region with boundaries at \(\pm r\), \(r>0\). Conditions are known for \(X\) to be in the Feller class \(FC_0\) at zero, by which we mean that each sequence \(t_k\downarrow 0\) contains a subsequence through which \(X(t_k)\), after norming by a nonstochastic function, converges to an a.s. finite nondegenerate random variable. We use these conditions on \(X\) to characterise similar properties for the normed positions \(\left \vert X(T( r))\right \vert /r\) and \(\left \vert X(T( r) -)\right \vert /r\), and also for the normed jump \(\left \vert \Delta X(T(r))/r\right \vert = \left \vert X(T(r))-X(T(r)-)\right \vert /r\) (“the jump causing ruin”), as convergence takes place through sequences \(r_k\downarrow 0\). We go on to give conditions for the continuity of distributions of the limiting random variables obtained in this way.

MSC:

62E17	Approximations to statistical distributions (nonasymptotic)
62B15	Theory of statistical experiments
62G05	Nonparametric estimation
60G65	Nonlinear processes (e.g., \(G\)-Brownian motion, \(G\)-Lévy processes)

Keywords:

Lévy process; passage time distributions; two-sided exit problem; domain of partial attraction; generalised Feller class; stochastic compactness

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