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

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.


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)
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[1] Barndorff-Nielsen, O.E. and Shephard, N. (2001) Modelling by Lévy processes. Selected Proceedings of the Symposium on Inference for Stochastic Processes (Athens, GA, 2000), 25-31, IMS Lecture Notes Monogr. Ser., 37, Inst. Math. Statist., Beachwood, OH.
[2] Bertoin, J. (1996) Lévy Processes. Cambridge Tracts in Mathematics, 121. Cambridge University Press, Cambridge.
[3] Bickel, P.J. and Doksum, K.A. (2015) Mathematical Statistics-Basic Ideas and Selected Topics. Vol. 1. 2nd Ed. Texts in Statistical Science Series. CRC Press, Boca Raton, FL, 2015. · Zbl 1380.62002
[4] Billingsley, P. (1999) Convergence of Probability Measures, Wiley series in probability and statistics, 2nd Ed. · Zbl 0944.60003
[5] Buchmann, B., Maller, R.A. and Mason, D.M. (2015) Laws of the iterated logarithm for self-normalised Lévy processes at zero. Trans. Amer. Math. Soc. 367, 1737-1770. · Zbl 1309.60024
[6] de Haan, L. and Ridder, G. (1979) Stochastic compactness of sample extremes. Ann. Probab., 7, 290-303. · Zbl 0395.60029
[7] Doney, R.A. and Maller, R.A. (2002) Stability of the overshoot for Lévy processes. Ann. Probab. 30, 188-212. · Zbl 1016.60052
[8] Erickson, K. and Maller, R. (2007) Finiteness of integrals of functions of Lévy processes, Proc. London Math. Soc., 94, 386-420. · Zbl 1121.60057
[9] Feller, W. (1965-66) On regular variation and local limit theorems, Proc. V Berkeley Symp. Math. Stats. Prob. II, Part I, 373-388.
[10] Griffin, P.S. and Maller, R.A. (1999) On compactness properties of the exit position of a random walk from an interval. Proc. London Math. Soc., 78, 459-480. · Zbl 1027.60041
[11] Griffin, P.S. and McConnell, T.R. (1992) On the position of a random walk at the time of first exit from a sphere, Ann. Prob., 20, 825-854. · Zbl 0756.60060
[12] Griffin, P.S. and McConnell, T.R. (1994) Gambler’s ruin and the first exit position of random walk from large spheres. Ann. Prob. 22, 1429-1472. · Zbl 0820.60055
[13] Kallenberg, O. (2002) Foundations of Modern Probability, Springer, New York, Berlin, Heidelberg. · Zbl 0996.60001
[14] Maller, R.A. (2009) Small-time versions of Strassen’s law for Lévy processes, Proc. London Math. Soc., 98, 531-558. · Zbl 1157.60044
[15] Maller, R.A. and Mason, D.M. (2010) Small-time compactness and convergence behaviour of deterministically and self-normalised Lévy processes, Trans. Amer. Math. Soc., 362, 2205-2248. · Zbl 1202.60036
[16] Maller, R.A. and Mason, D.M. (2013) A characterization of small and large time limit laws for self-normalized Lévy processes. In: Limit theorems in probability, statistics and number theory, 141-169, Springer Proc. Math. Stat., 42, Springer, Heidelberg. · Zbl 1277.60083
[17] Maller, R.A. and Mason, D.M. (2015) Matrix normalized convergence of a Lévy process to normality at zero, Stoch. Proc. Appl., 125, 2353-2382. · Zbl 1316.60068
[18] Maller, R.A. and Mason, D.M. (2018) Matrix normalised stochastic compactness for a Lévy process at zero. Electron. J. Probab. 23, Paper No. 69. · Zbl 1395.60053
[19] Pruitt, W.E. (1981) The growth of random walks and Lévy processes. Ann. Probab. 9, 948-956. · Zbl 0477.60033
[20] Roberts, A.W and Varberg, D.E. (1973) Convex functions. Pure and Applied Mathematics, Vol. 57. Academic Press. · Zbl 0271.26009
[21] Sato, Ken-iti (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics, 68. Cambridge University Press, Cambridge.
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