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BEJ 22 4.Pdf Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Two Number Four November 2016 ISSN: 1350-7265 CONTENTS Papers MALLER, R.A. 1963 Conditions for a Lévy process to stay positive near 0, in probability LACAUX, C. and SAMORODNITSKY, G. 1979 Time-changed extremal process as a random sup measure BISCIO, C.A.N. and LAVANCIER, F. 2001 Quantifying repulsiveness of determinantal point processes SHAO, Q.-M. and ZHOU, W.-X. 2029 Cramér type moderate deviation theorems for self-normalized processes WANG, M. and MARUYAMA, Y. 2080 Consistency of Bayes factor for nonnested model selection when the model dimension grows LANCONELLI, A. and STAN, A.I. 2101 A note on a local limit theorem for Wiener space valued random variables HUCKEMANN, S., KIM, K.-R., MUNK, A., REHFELDT, F., 2113 SOMMERFELD, M., WEICKERT, J. and WOLLNIK, C. The circular SiZer, inferred persistence of shape parameters and application to early stem cell differentiation DÖRING, H., FARAUD, G. and KÖNIG, W. 2143 Connection times in large ad-hoc mobile networks DELYON, B. and PORTIER, F. 2177 Integral approximation by kernel smoothing CORTINES, A. 2209 The genealogy of a solvable population model under selection with dynamics related to directed polymers ARNAUDON, M. and MICLO, L. 2237 A stochastic algorithm finding p-means on the circle HEAUKULANI, C. and ROY, D.M. 2301 The combinatorial structure of beta negative binomial processes (continued) The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Mathematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), and Zentralblatt für Mathematik (also avalaible on the MATH via STN database and Compact MATH CD-ROM). A list of forthcoming papers can be found online at http://www. bernoulli-society.org/index.php/publications/bernoulli-journal/bernoulli-journal-papers Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Two Number Four November 2016 ISSN: 1350-7265 CONTENTS (continued) Papers BUCHMANN, B., FAN, Y. and MALLER, R.A. 2325 Distributional representations and dominance of a Lévy process over its maximal jump processes THÄLE, C. and YUKICH, J.E. 2372 Asymptotic theory for statistics of the Poisson–Voronoi approximation PUPLINSKAITE,˙ D. and SURGAILIS, D. 2401 Aggregation of autoregressive random fields and anisotropic long-range dependence BALLY, V. and CARAMELLINO, L. 2442 Asymptotic development for the CLT in total variation distance PERKOWSKI, N. and PRÖMEL, D.J. 2486 Pathwise stochastic integrals for model free finance KANIKA and KUMAR, S. 2521 Methods for improving estimators of truncated circular parameters BAYRAKTAR, E. and MUNK, A. 2548 An α-stable limit theorem under sublinear expectation DENISOV, D. and LEONENKO, N. 2579 Limit theorems for multifractal products of geometric stationary processes Author Index 2609 Bernoulli 22(4), 2016, 1963–1978 DOI: 10.3150/15-BEJ716 Conditions for a Lévy process to stay positive near 0, in probability ROSS A. MALLER School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT, Australia. E-mail: [email protected] A necessary and sufficient condition for a Lévy process X to stay positive, in probability, near 0, which arises in studies of Chung-type laws for X near 0, is given in terms of the characteristics of X. Keywords: Lévy process; staying positive References [1] Andrew, P. (2008). On the limiting behaviour of Lévy processes at zero. Probab. Theory Related Fields 140 103–127. MR2357672 [2] Aurzada, F., Döring, L. and Savov, M. (2013). Small time Chung-type LIL for Lévy processes. Bernoulli 19 115–136. MR3019488 [3] Bertoin, J., Doney, R.A. and Maller, R.A. (2008). Passage of Lévy processes across power law bound- aries at small times. Ann. Probab. 36 160–197. MR2370602 [4] Buchmann, B., Fan, Y. and Maller, R.A. (2015). Distributional representations and dominance of a Lévy process over its maximal jump processes. Bernoulli. To appear. Available at arXiv:1409.4050. [5] Doney, R.A. (2004). Small-time behaviour of Lévy processes. Electron. J. Probab. 9 209–229. MR2041833 [6] Doney, R.A. and Maller, R.A. (2002). Stability and attraction to normality for Lévy processes at zero and at infinity. J. Theoret. Probab. 15 751–792. MR1922446 [7] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications (New York). New York: Springer. MR1876169 [8] Kesten, H. and Maller, R.A. (1997). Divergence of a random walk through deterministic and random subsequences. J. Theoret. Probab. 10 395–427. MR1455151 [9] Sato, K. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge Univ. Press. [10] Wee, I.S. (1988). Lower functions for processes with stationary independent increments. Probab. The- ory Related Fields 77 551–566. MR0933989 1350-7265 © 2016 ISI/BS Bernoulli 22(4), 2016, 1979–2000 DOI: 10.3150/15-BEJ717 Time-changed extremal process as a random sup measure CÉLINE LACAUX1,2,3 and GENNADY SAMORODNITSKY4 1Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France. E-mail: [email protected] 2CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France 3Inria, BIGS, Villers-lès-Nancy, F-54600, France 4School of Operations Research and Information Engineering and Department of Statistical Science Cor- nell University, Ithaca, NY 14853, USA. E-mail: [email protected] A functional limit theorem for the partial maxima of a long memory stable sequence produces a limiting process that can be described as a β-power time change in the classical Fréchet extremal process, for β in a subinterval of the unit interval. Any such power time change in the extremal process for 0 <β<1 produces a process with stationary max-increments. This deceptively simple time change hides the much more del- icate structure of the resulting process as a self-affine random sup measure. We uncover this structure and show that in a certain range of the parameters this random measure arises as a limit of the partial maxima of the same long memory stable sequence, but in a different space. These results open a way to construct a whole new class of self-similar Fréchet processes with stationary max-increments. Keywords: extremal limit theorem; extremal process; heavy tails; random sup measure; stable process; stationary max-increments; self-similar process References [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. New York: Wiley. MR1700749 [2] Davis, R. and Resnick, S. (1985). Limit theory for moving averages of random variables with regularly varying tail probabilities. Ann. Probab. 13 179–195. MR0770636 [3] Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton Series in Applied Mathemat- ics. Princeton, NJ: Princeton Univ. Press. MR1920153 [4] Fasen, V. (2005). Extremes of regularly varying Lévy-driven mixed moving average processes. Adv. in Appl. Probab. 37 993–1014. MR2193993 [5] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. II.NewYork: Wiley. MR0210154 [6] Harris, T.E. and Robbins, H. (1953). Ergodic theory of Markov chains admitting an infinite invariant measure. Proc. Natl. Acad. Sci. USA 39 860–864. MR0056873 [7] Jacod, J. and Shiryaev, A.N. (1987). Limit Theorems for Stochastic Processes. Grundlehren der Math- ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Berlin: Springer. MR0959133 [8] Kyprianou, A.E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Berlin: Springer. MR2250061 [9] Lamperti, J. (1964). On extreme order statistics. Ann. Math. Statist. 35 1726–1737. MR0170371 1350-7265 © 2016 ISI/BS [10] Leadbetter, M.R. (1983). Extremes and local dependence in stationary sequences. Z. Wahrsch. Verw. Gebiete 65 291–306. MR0722133 [11] Leadbetter, M.R., Lindgren, G. and Rootzén, H. (1983). Extremes and Related Properties of Random Sequences and Processes. Springer Series in Statistics. New York: Springer. MR0691492 [12] Mikosch, T. and Staric˘ a,˘ C. (2000). Limit theory for the sample autocorrelations and extremes of a GARCH(1, 1) process. Ann. Statist. 28 1427–1451. MR1805791 [13] Molchanov, I. (2005). Theory of Random Sets. Probability and Its Applications (New York). London: Springer. MR2132405 [14] O’Brien, G.L., Torfs, P.J.J.F. and Vervaat, W. (1990). Stationary self-similar extremal processes. Probab. Theory Related Fields 87 97–119. MR1076958 [15] Owada, T. and Samorodnitsky, G. (2015). Maxima of long memory stationary symmetric α-stable processes, and self-similar processes with stationary max-increments. Bernoulli 21 1575–1599. MR3352054 [16] Owada, T. and Samorodnitsky, G. (2015). Functional central limit theorem for heavy tailed stationary infinitely divisible processes generated by conservative flows. Ann. Probab. 43 240–285. MR3298473 [17] Resnick, S., Samorodnitsky, G. and Xue, F. (2000). Growth rates of sample covariances of stationary symmetric α-stable processes associated with null recurrent Markov chains. Stochastic Process. Appl. 85 321–339. MR1731029 [18] Rootzén, H. (1978). Extremes of moving averages of stable processes. Ann. Probab. 6 847–869. MR0494450 [19] Samorodnitsky, G. (2004). Extreme value theory, ergodic theory and the boundary between short memory and long memory for stationary stable processes. Ann. Probab. 32 1438–1468. MR2060304 [20] Samorodnitsky, G. (2006). Long range dependence. Found.
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