Smalltime compactness and convergence behavior of deterministically and selfnormalised Lévy processes
Authors:
Ross Maller and David M. Mason
Journal:
Trans. Amer. Math. Soc. 362 (2010), 22052248
MSC (2000):
Primary 60F05, 60F17, 60G51
Published electronically:
November 18, 2009
MathSciNet review:
2574893
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Abstract: Consider a Lévy process with quadratic variation process , , where denotes the jump process of . We give stability and compactness results, as , for the convergence both of the deterministically normed (and possibly centered) processes and , as well as theorems concerning the ``selfnormalised'' process . Thus, we consider the stochastic compactness and convergence in distribution of the 2vector , for deterministic functions and , as , possibly through a subsequence; and the stochastic compactness and convergence in distribution of , possibly to a nonzero constant (for stability), as , again possibly through a subsequence. As a main application it is shown that , a standard normal random variable, as , if and only if , as , for some nonstochastic function ; thus, is in the domain of attraction of the normal distribution, as , with or without centering constants being necessary (these being equivalent). We cite simple analytic equivalences for the above properties, in terms of the Lévy measure of . Functional versions of the convergences are also given.
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 Beichelt, F. (2006) Stochastic Processes in Science, Engineering and Finance, CRC Press, Boca Raton, FL. MR 2207217
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 Bertoin, J. Lévy Processes. (1996) Cambridge University Press, Cambridge. MR 1406564 (98e:60117)
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 Bertoin, J., Doney, R.A., and Maller, R.A. (2008) Passage of Lévy Processes across Power Law Boundaries at Small Times, Ann. Probab., 36, 160197. MR 2370602 (2009d:60141)
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 Blumenthal, R.M. and Getoor, R.K. (1961) Sample functions of stochastic processes with stationary independent increments. J. Math. Mech., 10, 492516. MR 0123362 (23:A689)
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 Buchmann, B., Maller, R.A., and Szimayer, A. (2008) An almost sure functional limit theorem at zero for a Lévy process normed by the square root function, and applications, Prob. Theor. Rel. Fields, 142, 219247. MR 2413271
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 de la Peña, V.H., Lai, T.L. and Shao, Q.M. (2009) SelfNormalized Processes: Limit Theory and Statistical Applications. SpringerVerlag, Berlin
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 Doney, R.A. and Maller, R.A. (2002) Stability of the overshoot for Lévy processes, Ann. Prob., 30, 188212. MR 1894105 (2003d:60090)
 13.
 Doney, R. A. and Maller, R.A. (2005) Passage times of random walks and Lévy processes across power law boundaries. Prob. Theor. Rel. Fields. 133, 5770. MR 2197137 (2007f:60038)
 14.
 Erickson, K.B., and Maller, R.A. (2007) Finiteness of integrals of functions of Lévy processes, Proc. Lond. Math. Soc., 94, 386420. MR 2308232 (2008g:60140)
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 Feller, W. (1966) On regular variation and local limit theorems. In: Proc. Fifth Berkeley Symp. Math. Statist. Prob., 2, 373388. Univ. California Press, Berkeley. MR 0219117 (36:2200)
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 Giné, E., Götze, F., and Mason, D.M. (1997) When is the student statistic asymptotically standard normal? Ann. Prob. 25, 15141531. MR 1457629 (98j:60033)
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 Giné, E. and Mason, D.M. (1998) On the LIL for SelfNormalized Sums of IID Random Variables, J. Theoretical Probab., 11, 351370. MR 1622575 (99e:60082)
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 Madan, D.B. and Seneta, E. (1990) The Variance Gamma (V.G.) model for share market returns. J. Business, 63, 511524.
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 Maller, R.A. (1981) Some properties of stochastic compactness. J. Austral. Math. Soc., 30, 264277. MR 614077 (82h:60043)
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 Maller, R.A. (1981) A theorem on products of random variables with application to regression. Austral. J. Statist., 23, 177185. MR 636133 (82m:60032)
 34.
 Maller, R.A. and Mason, D.M. (2008) Convergence in distribution of Lévy processes at small times with selfnormalisation, Acta Sci. Math. (Szeged), 74, 315347. MR 2431109
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 Mason, D.M. (2005) The asymptotic distribution of selfnormalized triangular arrays. J. Theor. Probab., 18, 853870 MR 2289935 (2008m:60035)
 36.
 Mason, D.M. and Zinn, J. (2005) When does a randomly weighted selfnormalized sum converge in distribution? Electronic Comm. Prob., 10, 7081. MR 2133894 (2005m:60045)
 37.
 Pruitt, W.E. (1981) The growth of random walks and Lé vy processes. Ann. Probab. 9, 948956. MR 632968 (84h:60063)
 38.
 Raikov, D.A., (1938) On a connection between the central limit theorem in the theory of probability and the law of large numbers. Izvestiya Akad. Nauk SSSR Ser. Mat., 323338.
 39.
 Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge. MR 1739520 (2003b:60064)
 40.
 Woerner, J. (2007) Inference in Lévytype stochastic volatility models, Adv. Appl. Prob. 39, 531549. MR 2343676
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Additional Information
Ross Maller
Affiliation:
Centre for Mathematical Analysis & School of Finance and Applied Statistics, Australian National University, PO Canberra, ACT, Australia
Email:
Ross.Maller@anu.edu.au
David M. Mason
Affiliation:
Food and Resource Economics, University of Delaware, 206 Townsend Hall, Newark, Delaware 19717
Email:
davidm@Udel.Edu
DOI:
http://dx.doi.org/10.1090/S0002994709050326
Received by editor(s):
June 10, 2008
Received by editor(s) in revised form:
March 3, 2009
Published electronically:
November 18, 2009
Additional Notes:
The first author’s research was partially supported by ARC Grant DP0664603
The second author’s research was partially supported by NSF Grant DMS–0503908.
Article copyright:
© Copyright 2009
American Mathematical Society
