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Survival probabilities of autoregressive processes

Published online by Cambridge University Press:  28 November 2013

Christoph Baumgarten
Christoph Baumgarten*
Affiliation:
Technische Universität Braunschweig, Institut für Mathematische Stochastik, Pockelsstrasse 14, 38106 Braunschweig, Germany. baumgart@math.tu-berlin.de
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Abstract

Given an autoregressive process X of order p (i.e. Xn = a1Xn−1 + ··· + apXn−p + Yn where the random variables Y1, Y2,... are i.i.d.), we study the asymptotic behaviour of the probability that the process does not exceed a constant barrier up to time N (survival or persistence probability). Depending on the coefficients a1,..., ap and the distribution of Y1, we state conditions under which the survival probability decays polynomially, faster than polynomially or converges to a positive constant. Special emphasis is put on AR(2) processes.

Keywords

Autoregressive processautoregressive moving averageboundary crossing probabilityone-sided exit problempersistence probablitysurvival probability
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 145 - 170
Copyright
© EDP Sciences, SMAI, 2013

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References

Aurzada, F. and Baumgarten, C., Survival probabilities of weighted random walks. ALEA Lat. Amer. J. Probab. Math. Stat. 8 (2011) 235–258. Google Scholar
F. Aurzada and T. Simon, Persistence probabilities and exponents. arXiv:1203.6554 (2012).
P.J. Brockwell and R.A. Davis, Time series: theory and methods. Springer Series in Statistics. Springer-Verlag, New York (1987).
A. Dembo, J. Ding and F. Gao, Persistence of iterated partial sums. Ann. Inst. Henri Poincaré B. To appear (2012).
Dembo, A., Poonen, B., Shao, Q.-M. and Zeitouni, O., Random polynomials having few or no real zeros. J. Amer. Math. Soc. 15 857–892 (2002). Electronic. Google Scholar
Doney, R.A., On the asymptotic behaviour of first passage times for transient random walk. Probab. Theory Related Fields 81 (1989) 239–246,. Google Scholar
S.N. Elaydi, An introduction to difference equations. Undergraduate Texts in Mathematics. Second edition, Springer-Verlag, New York (1999).
Esary, J.D., Proschan, F. and Walkup, D.W., Association of random variables, with applications. Ann. Math. Statist. 38 (1967) 1466–1474. Google Scholar
W. Feller, An introduction to probability theory and its applications. Second edition, John Wiley and Sons Inc., New York (1971).
G.R. Grimmett and D.R. Stirzaker, One thousand exercises in probability. Oxford University Press, Oxford (2001).
M. Ledoux and M. Talagrand, Probability in Banach spaces. Springer-Verlag, Berlin Heidelberg New York (1991).
Li, W.V. and Shao, Q.-M., Recent developments on lower tail probabilities for Gaussian processes. Cosmos 1 (2005) 95–106. Google Scholar
E. Lukacs, Characteristic functions. Second edition, revised and enlarged. Hafner Publishing Co., New York (1970).
Novikov, A. and Kordzakhia, N., Martingales and first passage times of AR(1) sequences. Stochast. 80 (2008) 197–210. Google Scholar
Y. Peres, W. Schlag and B. Solomyak, Sixty years of Bernoulli convolutions. In Fractal geometry and stochastics, II (Greifswald/Koserow, 1998), vol. 46 of Progr. Probab. Birkhäuser, Basel (2000) 39–65.
Sinaĭ, Ya. G., Statistics of shocks in solutions of inviscid Burgers equation. Commun. Math. Phys. 148 (1992) 601–621. Google Scholar