Official Journal of the Bernoulli Society for Mathematical Statisticsand Probability

Volume Twenty Two Number Four November 2016 ISSN: 1350-7265

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 andCompact 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

CONTENTS

Papers1963MALLER, R.A.

Conditions for a Lévy process to stay positive near 0, in probability

1979LACAUX, C. and SAMORODNITSKY, G.Time-changed extremal process as a random sup measure

2001BISCIO, C.A.N. and LAVANCIER, F.Quantifying repulsiveness of determinantal point processes

2029SHAO, Q.-M. and ZHOU, W.-X.Cramér type moderate deviation theorems for self-normalized processes

2080WANG, M. and MARUYAMA, Y.Consistency of Bayes factor for nonnested model selection when the model dimensiongrows

2101LANCONELLI, A. and STAN, A.I.A note on a local limit theorem for Wiener space valued random variables

2113HUCKEMANN, S., KIM, K.-R., MUNK, A., REHFELDT, F.,SOMMERFELD, M., WEICKERT, J. and WOLLNIK, C.The circular SiZer, inferred persistence of shape parameters and application to early stemcell differentiation

2143DÖRING, H., FARAUD, G. and KÖNIG, W.Connection times in large ad-hoc mobile networks

2177DELYON, B. and PORTIER, F.Integral approximation by kernel smoothing

2209CORTINES, A.The genealogy of a solvable population model under selection with dynamics related todirected polymers

2237ARNAUDON, M. and MICLO, L.A stochastic algorithm finding p-means on the circle

2301HEAUKULANI, C. and ROY, D.M.The combinatorial structure of beta negative binomial processes

(continued)

Official Journal of the Bernoulli Society for Mathematical Statisticsand Probability

Volume Twenty Two Number Four November 2016 ISSN: 1350-7265

CONTENTS

(continued)

Papers2325BUCHMANN, B., FAN, Y. and MALLER, R.A.

Distributional representations and dominance of a Lévy process over its maximal jumpprocesses

2372THÄLE, C. and YUKICH, J.E.Asymptotic theory for statistics of the Poisson–Voronoi approximation

2401PUPLINSKAITE, D. and SURGAILIS, D.Aggregation of autoregressive random fields and anisotropic long-range dependence

2442BALLY, V. and CARAMELLINO, L.Asymptotic development for the CLT in total variation distance

2486PERKOWSKI, N. and PRÖMEL, D.J.Pathwise stochastic integrals for model free finance

2521KANIKA and KUMAR, S.Methods for improving estimators of truncated circular parameters

2548BAYRAKTAR, E. and MUNK, A.An α-stable limit theorem under sublinear expectation

2579DENISOV, D. and LEONENKO, N.Limit theorems for multifractal products of geometric stationary processes

2609Author Index

Bernoulli 22(4), 2016, 1963–1978DOI: 10.3150/15-BEJ716

Conditions for a Lévy process to stay positivenear 0, in probabilityROSS A. MALLER

School of Finance, Actuarial Studies and Statistics, Australian National University, Canberra, ACT,Australia. E-mail: Ross.Maller@anu.edu.au

A necessary and sufficient condition for a Lévy process X to stay positive, in probability, near 0, whicharises 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

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1350-7265 © 2016 ISI/BS

Bernoulli 22(4), 2016, 1979–2000DOI: 10.3150/15-BEJ717

Time-changed extremal process as a randomsup measureCÉLINE LACAUX1,2,3 and GENNADY SAMORODNITSKY4

1Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506,France. E-mail: Celine.Lacaux@univ-lorraine.fr2CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France3Inria, BIGS, Villers-lès-Nancy, F-54600, France4School of Operations Research and Information Engineering and Department of Statistical Science Cor-nell University, Ithaca, NY 14853, USA. E-mail: gs18@cornell.edu

A functional limit theorem for the partial maxima of a long memory stable sequence produces a limitingprocess that can be described as a β-power time change in the classical Fréchet extremal process, for β in asubinterval of the unit interval. Any such power time change in the extremal process for 0 < β < 1 producesa 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 andshow that in a certain range of the parameters this random measure arises as a limit of the partial maximaof the same long memory stable sequence, but in a different space. These results open a way to construct awhole 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

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Bernoulli 22(4), 2016, 2001–2028DOI: 10.3150/15-BEJ718

Quantifying repulsiveness of determinantalpoint processesCHRISTOPHE ANGE NAPOLÉON BISCIO1 and FRÉDÉRIC LAVANCIER2

1Laboratoire de Mathématiques Jean Leray – BP 92208 – 2, Rue de la Houssinière – F-44322 NantesCedex 03 – France. E-mail: Christophe.Biscio@univ-nantes.fr2Inria, Centre Rennes Bretagne Atlantique, France. E-mail: Frederic.Lavancier@univ-nantes.fr

Determinantal point processes (DPPs) have recently proved to be a useful class of models in several areas ofstatistics, including spatial statistics, statistical learning and telecommunications networks. They are modelsfor repulsive (or regular, or inhibitive) point processes, in the sense that nearby points of the process tendto repel each other. We consider two ways to quantify the repulsiveness of a point process, both basedon its second-order properties, and we address the question of how repulsive a stationary DPP can be.We determine the most repulsive stationary DPP, when the intensity is fixed, and for a given R > 0 weinvestigate repulsiveness in the subclass of R-dependent stationary DPPs, that is, stationary DPPs withR-compactly supported kernels. Finally, in both the general case and the R-dependent case, we presentsome new parametric families of stationary DPPs that can cover a large range of DPPs, from the stationaryPoisson process (the case of no interaction) to the most repulsive DPP.

Keywords: compactly supported covariance function; covariance function; pair correlation function;R-dependent point process

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Bernoulli 22(4), 2016, 2029–2079DOI: 10.3150/15-BEJ719

Cramér type moderate deviation theorems forself-normalized processesQI-MAN SHAO1 and WEN-XIN ZHOU2,3

1Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.E-mail: qmshao@cuhk.edu.hk2Department of Operations Research and Financial Engineering, Princeton University, Princeton,NJ 08544, USA. E-mail: wenxinz@princeton.edu3School of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia

Cramér type moderate deviation theorems quantify the accuracy of the relative error of the normal ap-proximation and provide theoretical justifications for many commonly used methods in statistics. In thispaper, we develop a new randomized concentration inequality and establish a Cramér type moderate devia-tion theorem for general self-normalized processes which include many well-known Studentized nonlinearstatistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is establishedfor Studentized U -statistics.

Keywords: moderate deviation; nonlinear statistics; relative error; self-normalized processes; Studentizedstatistics; U -statistics

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Bernoulli 22(4), 2016, 2080–2100DOI: 10.3150/15-BEJ720

Consistency of Bayes factor for nonnestedmodel selection when the modeldimension growsMIN WANG1 and YUZO MARUYAMA2

1Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA.E-mail: minwang@mtu.edu2Center for Spatial Information Science, University of Tokyo, Bunkyo-ku, Tokyo, 113-0033, Japan.E-mail: maruyama@csis.u-tokyo.ac.jp

Zellner’s g-prior is a popular prior choice for the model selection problems in the context of normal re-gression models. Wang and Sun [J. Statist. Plann. Inference 147 (2014) 95–105] recently adopt this priorand put a special hyper-prior for g, which results in a closed-form expression of Bayes factor for nestedlinear model comparisons. They have shown that under very general conditions, the Bayes factor is consis-tent when two competing models are of order O(nτ ) for τ < 1 and for τ = 1 is almost consistent excepta small inconsistency region around the null hypothesis. In this paper, we study Bayes factor consistencyfor nonnested linear models with a growing number of parameters. Some of the proposed results generalizethe ones of the Bayes factor for the case of nested linear models. Specifically, we compare the asymptoticbehaviors between the proposed Bayes factor and the intrinsic Bayes factor in the literature.

Keywords: Bayes factor; growing number of parameters; model selection consistency; nonnested linearmodels; Zellner’s g-prior

References

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[3] Casella, G., Girón, F.J., Martínez, M.L. and Moreno, E. (2009). Consistency of Bayesian proceduresfor variable selection. Ann. Statist. 37 1207–1228. MR2509072

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[8] Girón, F.J., Moreno, E., Casella, G. and Martínez, M.L. (2010). Consistency of objective Bayes factorsfor nonnested linear models and increasing model dimension. Rev. R. Acad. Cienc. Exactas FíS. Nat.Ser. A Math. RACSAM 104 57–67. MR2666441

1350-7265 © 2016 ISI/BS

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[10] Hoel, P.G. (1947). On the choice of forecasting formulas. J. Amer. Statist. Assoc. 42 605–611.[11] Kass, R.E. and Raftery, A.E. (1995). Bayes factors. J. Amer. Statist. Assoc. 90 773–795.[12] Kass, R.E. and Vaidyanathan, S.K. (1992). Approximate Bayes factors and orthogonal parameters,

with application to testing equality of two binomial proportions. J. Roy. Statist. Soc. Ser. B 54 129–144. MR1157716

[13] Ley, E. and Steel, M.F.J. (2012). Mixtures of g-priors for Bayesian model averaging with economicapplications. J. Econometrics 171 251–266. MR2991863

[14] Liang, F., Paulo, R., Molina, G., Clyde, M.A. and Berger, J.O. (2008). Mixtures of g priors forBayesian variable selection. J. Amer. Statist. Assoc. 103 410–423. MR2420243

[15] Maruyama, Y. (2013). A Bayes factor with reasonable model selection consistency for ANOVA model.Available at arXiv:0906.4329v2 [stat.ME].

[16] Maruyama, Y. and George, E.I. (2011). Fully Bayes factors with a generalized g-prior. Ann. Statist.39 2740–2765. MR2906885

[17] Moreno, E. and Girón, F.J. (2008). Comparison of Bayesian objective procedures for variable selectionin linear regression. TEST 17 472–490. MR2470092

[18] Moreno, E., Girón, F.J. and Casella, G. (2010). Consistency of objective Bayes factors as the modeldimension grows. Ann. Statist. 38 1937–1952. MR2676879

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Bernoulli 22(4), 2016, 2101–2112DOI: 10.3150/15-BEJ721

A note on a local limit theorem for Wienerspace valued random variablesALBERTO LANCONELLI1 and AUREL I. STAN2

1Dipartimento di Matematica, Universitá degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari,Italia. E-mail: alberto.lanconelli@uniba.it2Department of Mathematics, Ohio State University at Marion, 1465 Mount Vernon Avenue, Marion,OH 43302, USA. E-mail: stan.7@osu.edu

We prove a local limit theorem, that is, a central limit theorem for densities, for a sequence of independentand identically distributed random variables taking values on an abstract Wiener space; the common lawof those random variables is assumed to be absolutely continuous with respect to the reference Gaussianmeasure. We begin by showing that the key roles of scaling operator and convolution product in this infinitedimensional Gaussian framework are played by the Ornstein–Uhlenbeck semigroup and Wick product,respectively. We proceed by establishing a necessary condition on the density of the random variablesfor the local limit theorem to hold true. We then reverse the implication and prove under an additionalassumption the desired L1-convergence of the density of X1+···+Xn√

n. We close the paper comparing our

result with certain Berry–Esseen bounds for multidimensional central limit theorems.

Keywords: abstract Wiener space; local limit theorem; Ornstein–Uhlenbeck semigroup; Wick product

References

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Gaussian Wick products. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 375–407. MR2847245[7] Da Pelo, P., Lanconelli, A. and Stan, A.I. (2013). An Itô formula for a family of stochastic integrals

and related Wong–Zakai theorems. Stochastic Process. Appl. 123 3183–3200. MR3062442[8] Gnedenko, B.V. (1954). A local limit theorem for densities. Dokl. Akad. Nauk SSSR 95 5–7.

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[11] Lanconelli, A. and Sportelli, L. (2012). Wick calculus for the square of a Gaussian random variablewith application to Young and hypercontractive inequalities. Infin. Dimens. Anal. Quantum Probab.Relat. Top. 15 1250018, 16. MR2999099

[12] Lanconelli, A. and Stan, A.I. (2010). Some norm inequalities for Gaussian Wick products. Stoch.Anal. Appl. 28 523–539. MR2739573

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MR0131887

Bernoulli 22(4), 2016, 2113–2142DOI: 10.3150/15-BEJ722

The circular SiZer, inferred persistence ofshape parameters and application to earlystem cell differentiationSTEPHAN HUCKEMANN1,*, KWANG-RAE KIM2,** , AXEL MUNK3,† ,FLORIAN REHFELDT4,‡, MAX SOMMERFELD1,§,JOACHIM WEICKERT5,¶ and CARINA WOLLNIK4,‖1Felix Bernstein Institute for Mathematical Statistics in the Biosciences, University of Göttingen.E-mail: *huckeman@math.uni-goettingen.de; §max.sommerfeld@math.uni-goettingen.de2School of Mathematical Sciences, University of Nottingham. E-mail: **Kwang-rae.Kim@nottingham.ac.uk3Max Planck Institute for Biophysical Chemistry, Göttingen and Felix Bernstein Institute for MathematicalStatistics in the Biosciences, University of Göttingen. E-mail: †munk@math.uni-goettingen.de43rd Institute of Physics – Biophysics, University of Göttingen.E-mail: ‡rehfeldt@physik3.gwdg.de; ‖carina.wollnik@phys.uni-goettingen.de5Faculty of Mathematics and Computer Science, Saarland University.E-mail: ¶weickert@mia.uni-saarland.de

We generalize the SiZer of Chaudhuri and Marron (J. Amer. Statist. Assoc. 94 (1999) 807–823; Ann. Statist.28 (2000) 408–428) for the detection of shape parameters of densities on the real line to the case of circulardata. It turns out that only the wrapped Gaussian kernel gives a symmetric, strongly Lipschitz semi-groupsatisfying “circular” causality, that is, not introducing possibly artificial modes with increasing levels ofsmoothing. Some notable differences between Euclidean and circular scale space theory are highlighted.Based on this, we provide an asymptotic theory to make inference about the persistence of shape features.The resulting circular mode persistence diagram is applied to the analysis of early mechanically-induceddifferentiation in adult human stem cells from their actin-myosin filament structure. As a consequence, thecircular SiZer based on the wrapped Gaussian kernel (WiZer) allows the verification at a controlled errorlevel of the observation reported by Zemel et al. (Nat. Phys. 6 (2010) 468–473): Within early stem celldifferentiation, polarizations of stem cells exhibit preferred directions in three different micro-environments.

Keywords: circular data; circular scale spaces; mode hunting; multiscale process; persistence inference;stem cell differentiation; variation diminishing; wrapped Gaussian kernel estimator

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Bernoulli 22(4), 2016, 2143–2176DOI: 10.3150/15-BEJ724

Connection times in large ad-hocmobile networksHANNA DÖRING1, GABRIEL FARAUD2 and WOLFGANG KÖNIG3,4

1Universität Osnabrück, Institut für Mathematik, Albrechtstr. 28a, 49076 Osnabrück, Germany.E-mail: hanna.doering@uos.de2Laboratoire Modal’x, Université Paris 10 Nanterre-La Défense, 200 Av. de la République, 92000 Nanterre,France. E-mail: gabriel.faraud@u-paris10.fr3Weierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germany. E-mail: koenig@wias-berlin.de4Technische Universität Berlin, Institut für Mathematik, Str. des 17. Juni 136, 10623 Berlin, Germany

We study connectivity properties in a probabilistic model for a large mobile ad-hoc network. We considera large number of participants of the system moving randomly, independently and identically distributedin a large domain, with a space-dependent population density of finite, positive order and with a fixedtime horizon. Messages are instantly transmitted according to a relay principle, that is, they are iterativelyforwarded from participant to participant over distances smaller than the communication radius until theyreach the recipient. In mathematical terms, this is a dynamic continuum percolation model.

We consider the connection time of two sample participants, the amount of time over which these twoare connected with each other. In the above thermodynamic limit, we find that the connectivity inducedby the system can be described in terms of the counterplay of a local, random and a global, deterministicmechanism, and we give a formula for the limiting behaviour.

A prime example of the movement schemes that we consider is the well-known random waypoint model.Here, we give a negative upper bound for the decay rate, in the limit of large time horizons, of the probabilityof the event that the portion of the connection time is less than the expectation.

Keywords: ad-hoc networks; connectivity; dynamic continuum percolation; large deviations; randomwaypoint model

References

[1] Asmussen, S. (2003). Applied Probability and Queues, 2nd ed. Applications of Mathematics (NewYork) 51. New York: Springer. MR1978607

[2] Bettstetter, C. and Wagner, C. (2002). The spatial node distribution of the random waypoint mobilitymodel. WMAN, 41–58.

[3] Bettstetter, C., Hartenstein, H. and Pérez-Costa, X. and (2004). Stochastic properties of the randomwaypoint mobility model. ACM/Kluwer Wireless Networks 6, Special Issue on Modeling and Analysisof Mobile Networks 10 555–567.

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[5] Camp, T., Boleng, J. and Davies, V. (2002). A survey of mobility models for ad-hoc network researchWCMC: Special Issue on Mobile ad-hoc Networking: Research, Trends and Applications 2 483–502.

[6] Clementi, A.E.F., Pasquale, F. and Silvestri, R. (2009). MANETS: High mobility can make up for lowtransmission power. In Automata, Languages and Programming. Part II. Lecture Notes in ComputerScience 5556 387–398. Berlin: Springer. MR2544811

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[7] Dembo, A. and Zeitouni, O. (2010). Large Deviations Techniques and Applications. Stochastic Mod-elling and Applied Probability 38. Berlin: Springer. Corrected reprint of the second (1998) edition.MR2571413

[8] Hyytia, E., Lassila, P. and Virtamo, J. (2006). Spatial node distribution of the random waypoint mo-bility model with applications. IEEE Trans. Mob. Comput. 5 680–694.

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[11] Le Boudec, J.-Y. (2007). Understanding the simulation of mobility models with palm calculus. Per-form. Eval. 64 126–146.

[12] Le Boudec, J.-Y. and Vojnovic, M. (2006). The random trip model: Stability, stationary regime, andperfect simulation. IEEE/ACM Transactions on Networking 14 1153–1166.

[13] Meester, R. and Roy, R. (1996). Continuum Percolation. Cambridge Tracts in Mathematics 119. Cam-bridge: Cambridge Univ. Press. MR1409145

[14] Penrose, M.D. (2003). Random Geometric Graphs. Oxford Studies in Probability 5. Oxford: OxfordUniv. Press. MR1986198

[15] Penrose, M.D. (1991). On a continuum percolation model. Adv. in Appl. Probab. 23 536–556.MR1122874

[16] Penrose, M.D. (1995). Single linkage clustering and continuum percolation. J. Multivariate Anal. 5394–109. MR1333129

[17] Peres, Y., Sinclair, A., Sousi, P. and Stauffer, A. (2013). Mobile geometric graphs: Detection, coverageand percolation. Probab. Theory Related Fields 156 273–305. MR3055260

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J. Appl. Probab. 34 363–371. MR1447341

Bernoulli 22(4), 2016, 2177–2208DOI: 10.3150/15-BEJ725

Integral approximation by kernel smoothingBERNARD DELYON1 and FRANÇOIS PORTIER2

1Institut de recherches mathématiques de Rennes (IRMAR), Campus de Beaulieu, Université de Rennes 1,35042 Rennes Cédex, France. E-mail: bernard.delyon@univ-rennes1.fr2Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain,Belgique. E-mail: francois.portier@gmail.com

Let (X1, . . . ,Xn) be an i.i.d. sequence of random variables in Rd , d ≥ 1. We show that, for any function

ϕ : Rd →R, under regularity conditions,

n1/2

(n−1

n∑i=1

ϕ(Xi)

f (Xi)−

∫ϕ(x) dx

)P−→ 0,

where f is the classical kernel estimator of the density of X1. This result is striking because it speedsup traditional rates, in root n, derived from the central limit theorem when f = f . Although this paperhighlights some applications, we mainly address theoretical issues related to the later result. We deriveupper bounds for the rate of convergence in probability. These bounds depend on the regularity of thefunctions ϕ and f , the dimension d and the bandwidth of the kernel estimator f . Moreover, they areshown to be accurate since they are used as renormalizing sequences in two central limit theorems eachreflecting different degrees of smoothness of ϕ. As an application to regression modelling with randomdesign, we provide the asymptotic normality of the estimation of the linear functionals of a regressionfunction. As a consequence of the above result, the asymptotic variance does not depend on the regressionfunction. Finally, we debate the choice of the bandwidth for integral approximation and we highlight thegood behavior of our procedure through simulations.

Keywords: central limit theorem; integral approximation; kernel smoothing; nonparametric regression

References

[1] Bickel, P.J., Klaassen, C.A.J., Ritov, Y. and Wellner, J.A. (1993). Efficient and Adaptive Estimation forSemiparametric Models. Johns Hopkins Series in the Mathematical Sciences. Baltimore, MD: JohnsHopkins Univ. Press. MR1245941

[2] Boucheron, S., Lugosi, G. and Bousquet, O. (2004). Concentration inequalities. In Advanced Lectureson Machine Learning. Lecture Notes in Computer Science 3176 208–240. Berlin: Springer.

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[6] Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596.MR0615434

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[7] Evans, L.C. and Gariepy, R.F. (1992). Measure Theory and Fine Properties of Functions. Studies inAdvanced Mathematics. Boca Raton, FL: CRC Press. MR1158660

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[10] Folland, G.B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Pure andApplied Mathematics (New York). New York: Wiley. MR1681462

[11] Gamboa, F., Loubes, J.-M. and Maza, E. (2007). Semi-parametric estimation of shifts. Electron. J.Stat. 1 616–640. MR2369028

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[13] Hall, P. and Heyde, C.C. (1980). Martingale Limit Theory and Its Application: Probability and Math-ematical Statistics. New York: Academic Press. MR0624435

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[15] Härdle, W., Marron, J.S. and Tsybakov, A.B. (1992). Bandwidth choice for average derivative estima-tion. J. Amer. Statist. Assoc. 87 218–226. MR1158640

[16] Härdle, W. and Stoker, T.M. (1989). Investigating smooth multiple regression by the method of aver-age derivatives. J. Amer. Statist. Assoc. 84 986–995. MR1134488

[17] Jones, M.C. (1993). Simple boundary correction for kernel density estimation. Stat. Comput. 3 135–146.

[18] Oh, M.-S. and Berger, J.O. (1992). Adaptive importance sampling in Monte Carlo integration. J. Stat.Comput. Simul. 41 143–168. MR1276184

[19] Robinson, P.M. (1988). Root-N -consistent semiparametric regression. Econometrica 56 931–954.MR0951762

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[23] Tsybakov, A.B. (2009). Introduction to Nonparametric Estimation. Springer Series in Statistics. NewYork: Springer. MR2724359

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[25] Zhang, P. (1996). Nonparametric importance sampling. J. Amer. Statist. Assoc. 91 1245–1253.MR1424622

Bernoulli 22(4), 2016, 2209–2236DOI: 10.3150/15-BEJ726

The genealogy of a solvable populationmodel under selection with dynamics relatedto directed polymersASER CORTINES

Université Paris Diderot, Mathématiques, case 7012, F-75 205 Paris Cedex 13, France.E-mail: cortines@math.univ-paris-diderot.fr

We consider a stochastic model describing a constant size N population that may be seen as a directedpolymer in random medium with N sites in the transverse direction. The population dynamics is governedby a noisy traveling wave equation describing the evolution of the individual fitnesses. We show that undersuitable conditions the generations are independent and the model is characterized by an extended Wright–Fisher model, in which the individual i has a random fitness ηi and the joint distribution of offspring(ν1, . . . , νN ) is given by a multinomial law with N trials and probability outcomes ηi ’s. We then showthat the average coalescence times scales like logN and that the limit genealogical trees are governedby the Bolthausen–Sznitman coalescent, which validates the predictions by Brunet, Derrida, Mueller andMunier for this class of models. We also study the extended Wright–Fisher model, and show that, undercertain conditions on ηi , the limit may be Kingman’s coalescent, a coalescent with multiple collisions, or acoalescent with simultaneous multiple collisions.

Keywords: ancestral processes; Bolthausen–Sznitman coalescent; coalescence; travelling waves

References

[1] Bender, C.M. and Orszag, S.A. (1999). Advanced Mathematical Methods for Scientists and Engineers.I: Asymptotic Methods and Perturbation Theory. New York: Springer. MR1721985

[2] Berestycki, J., Berestycki, N. and Schweinsberg, J. (2013). The genealogy of branching Brownianmotion with absorption. Ann. Probab. 41 527–618. MR3077519

[3] Bolthausen, E. and Sznitman, A.-S. (1998). On Ruelle’s probability cascades and an abstract cavitymethod. Comm. Math. Phys. 197 247–276. MR1652734

[4] Brunet, É. and Derrida, B. (2004). Exactly soluble noisy traveling-wave equation appearing in theproblem of directed polymers in a random medium. Phys. Rev. E (3) 70 016106, 5. MR2125704

[5] Brunet, É. and Derrida, B. (2013). Genealogies in simple models of evolution. J. Stat. Mech. TheoryExp. 1 P01006, 20. MR3036206

[6] Brunet, E., Derrida, B. and Damien, S. (2008). Universal tree structures in directed polymers andmodels of evolving populations. Phys. Rev. E 78 061102.

[7] Brunet, E., Derrida, B., Mueller, A.H. and Munier, S. (2006). Noisy traveling waves: Effect of selec-tion on genealogies. Europhys. Lett. 76 1–7. MR2299937

[8] Brunet, É., Derrida, B., Mueller, A.H. and Munier, S. (2007). Effect of selection on ancestry: Anexactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76 041104, 20.MR2365627

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[9] Comets, F., Quastel, J. and Ramírez, A.F. (2013). Last passage percolation and traveling fronts. J. Stat.Phys. 152 419–451. MR3082639

[10] Cook, J. and Derrida, B. (1990). Directed polymers in a random medium: 1/d expansion and then-tree approximation. J. Phys. A 23 1523–1554. MR1048783

[11] Cortines, A. (2014). Front velocity and directed polymers in random medium. Stochastic Process.Appl. 124 3698–3723. MR3249352

[12] Huillet, T. and Möhle, M. (2011). Population genetics models with skewed fertilities: A forward andbackward analysis. Stoch. Models 27 521–554. MR2827443

[13] Huillet, T. and Möhle, M. (2013). On the extended Moran model and its relation to coalescents withmultiple collisions. Theoretical Population Biology 87 5–14.

[14] Kingman, J.F.C. (1982). On the genealogy of large populations. J. Appl. Probab. 19A 27–43.MR0633178

[15] Möhle, M. (1999). Weak convergence to the coalescent in neutral population models. J. Appl. Probab.36 446–460. MR1724816

[16] Möhle, M. (2000). Total variation distances and rates of convergence for ancestral coalescent pro-cesses in exchangeable population models. Adv. in Appl. Probab. 32 983–993. MR1808909

[17] Möhle, M. and Sagitov, S. (2001). A classification of coalescent processes for haploid exchangeablepopulation models. Ann. Probab. 29 1547–1562. MR1880231

[18] Pitman, J. (1999). Coalescents with multiple collisions. Ann. Probab. 27 1870–1902. MR1742892[19] Schweinsberg, J. (2000). Coalescents with simultaneous multiple collisions. Electron. J. Probab. 5 50

pp. (electronic). MR1781024[20] Schweinsberg, J. (2003). Coalescent processes obtained from supercritical Galton–Watson processes.

Stochastic Process. Appl. 106 107–139. MR1983046

Bernoulli 22(4), 2016, 2237–2300DOI: 10.3150/15-BEJ728

A stochastic algorithm finding p-meanson the circleMARC ARNAUDON1 and LAURENT MICLO2

1Institut de Mathématique de Bordeaux, UMR 5251, Université de Bordeaux and CNRS, 351, Cours de laLibération, F-33405 TALENCE Cedex, France. E-mail: marc.arnaudon@u-bordeaux1.fr2Institut de Mathématiques de Toulouse, UMR 5219, Université Toulouse 3 and CNRS, 118, route de Nar-bonne, 31062 Toulouse Cedex 9, France. E-mail: miclo@math.univ-toulouse.fr

A stochastic algorithm is proposed, finding some elements from the set of intrinsic p-mean(s) associatedto a probability measure ν on a compact Riemannian manifold and to p ∈ [1,∞). It is fed sequentiallywith independent random variables (Yn)n∈N distributed according to ν, which is often the only availableknowledge of ν. Furthermore, the algorithm is easy to implement, because it evolves like a Brownian motionbetween the random times when it jumps in direction of one of the Yn, n ∈ N. Its principle is based onsimulated annealing and homogenization, so that temperature and approximations schemes must be tunedup (plus a regularizing scheme if ν does not admit a Hölderian density). The analysis of the convergence isrestricted to the case where the state space is a circle. In its principle, the proof relies on the investigation ofthe evolution of a time-inhomogeneous L2 functional and on the corresponding spectral gap estimates dueto Holley, Kusuoka and Stroock. But it requires new estimates on the discrepancies between the unknowninstantaneous invariant measures and some convenient Gibbs measures.

Keywords: Gibbs measures; homogenization; instantaneous invariant measures; intrinsic p-means;probability measures on compact Riemannian manifolds; simulated annealing; spectral gap at smalltemperature; stochastic algorithms

References

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[2] Arnaudon, M., Dombry, C., Phan, A. and Yang, L. (2012). Stochastic algorithms for computing meansof probability measures. Stochastic Process. Appl. 122 1437–1455. MR2914758

[3] Arnaudon, M. and Miclo, L. (2014). Means in complete manifolds: Uniqueness and approximation.ESAIM Probab. Stat. 18 185–206. MR3230874

[4] Arnaudon, M. and Miclo, L. (2014). A stochastic algorithm finding generalized means on compactmanifolds. Stochastic Process. Appl. 124 3463–3479. MR3231628

[5] Arnaudon, M. and Nielsen, F. (2012). Medians and means in Finsler geometry. LMS J. Comput. Math.15 23–37. MR2891143

[6] Badoiu, M. and Clarkson, K.L. (2003). Smaller core-sets for balls. In Proceedings of the FourteenthAnnual ACM-SIAM Symposium on Discrete Algorithms (Baltimore, MD, 2003) 801–802. New York:ACM. MR1974995

[7] Bonnabel, S. (2013). Stochastic gradient descent on Riemannian manifolds. IEEE Trans. Automat.Control 58 2217–2229. MR3101606

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[8] Cardot, H., Cénac, P. and Zitt, P.-A. (2013). Efficient and fast estimation of the geometric median inHilbert spaces with an averaged stochastic gradient algorithm. Bernoulli 19 18–43. MR3019484

[9] Catoni, O. (1999). Simulated annealing algorithms and Markov chains with rare transitions. In Sémi-naire de Probabilités, XXXIII. Lecture Notes in Math. 1709 69–119. Berlin: Springer. MR1767994

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[19] Jost, J. (2011). Riemannian Geometry and Geometric Analysis, 6th ed. Universitext. Heidelberg:Springer. MR2829653

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Toulouse Math. (6) 4 819–877. MR1623480[25] Miclo, L. (1996). Recuit simulé partiel. Stochastic Process. Appl. 65 281–298. MR1425361[26] Pennec, X. (2006). Intrinsic statistics on Riemannian manifolds: Basic tools for geometric measure-

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and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002). Contemp. Math. 338 357–390.Providence, RI: Amer. Math. Soc. MR2039961

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Bernoulli 22(4), 2016, 2301–2324DOI: 10.3150/15-BEJ729

The combinatorial structure of beta negativebinomial processesCREIGHTON HEAUKULANI1 and DANIEL M. ROY2

1Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, UnitedKingdom. E-mail: ckh28@cam.ac.uk2Department of Statistical Sciences, University of Toronto, 100 St. George Street, Toronto, ON, M5S 3G3,Canada. E-mail: droy@utstat.toronto.edu

We characterize the combinatorial structure of conditionally-i.i.d. sequences of negative binomial processeswith a common beta process base measure. In Bayesian nonparametric applications, such processes haveserved as models for latent multisets of features underlying data. Analogously, random subsets arise fromconditionally-i.i.d. sequences of Bernoulli processes with a common beta process base measure, in whichcase the combinatorial structure is described by the Indian buffet process. Our results give a count analogueof the Indian buffet process, which we call a negative binomial Indian buffet process. As an intermediatestep toward this goal, we provide a construction for the beta negative binomial process that avoids a repre-sentation of the underlying beta process base measure. We describe the key Markov kernels needed to usea NB-IBP representation in a Markov Chain Monte Carlo algorithm targeting a posterior distribution.

Keywords: Bayesian nonparametrics; Indian buffet process; latent feature models; multisets

References

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[4] Broderick, T., Pitman, J. and Jordan, M.I. (2013). Feature allocations, probability functions, and paint-boxes. Bayesian Anal. 8 801–836. MR3150470

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[7] Ghahramani, Z., Griffiths, T.L. and Sollich, P. (2007). Bayesian nonparametric latent feature models.In Bayesian Statistics 8. Oxford Sci. Publ. 201–226. Oxford: Oxford Univ. Press. MR2433194

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latent factors. In Advances in Neural Information Processing Systems 20, Vancouver, Canada.[18] Neal, R.M. (2003). Slice sampling. Ann. Statist. 31 705–767. With discussions and a rejoinder by the

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of the beta process. In Proceedings of the 27th International Conference on Machine Learning, Haifa,Israel.

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Bernoulli 22(4), 2016, 2325–2371DOI: 10.3150/15-BEJ731

Distributional representations and dominanceof a Lévy process over its maximal jumpprocessesBORIS BUCHMANN1,*, YUGUANG FAN2 and ROSS A. MALLER1,**

1Research School of Finance, Actuarial Studies & Statistics, Mathematical Sciences Institute, AustralianNational University, Australia. E-mail: *Boris.Buchmann@anu.edu.au; **Ross.Maller@anu.edu.au2School of Mathematics & Statistics, University of Melbourne, ARC Centre of Excellence for Mathematics& Statistical Frontiers, Australia. E-mail: Yuguang.Fan@unimelb.edu.au

Distributional identities for a Lévy process Xt , its quadratic variation process Vt and its maximal jump pro-cesses, are derived, and used to make “small time” (as t ↓ 0) asymptotic comparisons between them. Therepresentations are constructed using properties of the underlying Poisson point process of the jumps of X.Apart from providing insight into the connections between X, V , and their maximal jump processes, theyenable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised”versions of Xt , that is, Xt after division by sup0<s≤t �Xs , or by sup0<s≤t |�Xs |. Thus, we obtain nec-essary and sufficient conditions for Xt/ sup0<s≤t �Xs and Xt/ sup0<s≤t |�Xs | to converge in probabilityto 1, or to ∞, as t ↓ 0, so that X is either comparable to, or dominates, its largest jump. The former situationtends to occur when the singularity at 0 of the Lévy measure of X is fairly mild (its tail is slowly varyingat 0), while the latter situation is related to the relative stability or attraction to normality of X at 0 (a steepersingularity at 0). An important component in the analyses is the way the largest positive and negative jumpsinteract with each other. Analogous “large time” (as t → ∞) versions of the results can also be obtained.

Keywords: distributional representation; domain of attraction to normality; dominance; Lévy process;maximal jump process; relative stability

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[29] Khintchine, Y.A. (1937). Zur Theorie der unbeschränkt teilbaren Verteilungsgesetze. Mat. Sb. 2 79–119.

[30] Klass, M.J. and Wittmann, R. (1993). Which i.i.d. sums are recurrently dominated by their maximalterms? J. Theoret. Probab. 6 195–207. MR1215655

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Bernoulli 22(4), 2016, 2372–2400DOI: 10.3150/15-BEJ732

Asymptotic theory for statistics of thePoisson–Voronoi approximationCHRISTOPH THÄLE1 and J.E. YUKICH2

1Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany. E-mail: christoph.thaele@rub.de2Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA.E-mail: joseph.yukich@lehigh.edu

This paper establishes expectation and variance asymptotics for statistics of the Poisson–Voronoi approxi-mation of general sets, as the underlying intensity of the Poisson point process tends to infinity. Statisticsof interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensionalskeletons. We also consider the complexity of the so-called Voronoi zone and the iterated Voronoi approxi-mation. Our results are consequences of general limit theorems proved with an abstract Steiner-type formulaapplicable in the setting of sums of stabilizing functionals.

Keywords: combinatorial geometry; Poisson point process; Poisson–Voronoi approximation; randommosaic; stabilizing functional; stochastic geometry

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Bernoulli 13 1124–1150. MR2364229[22] Penrose, M.D. and Yukich, J.E. (2001). Central limit theorems for some graphs in computational

geometry. Ann. Appl. Probab. 11 1005–1041. MR1878288[23] Penrose, M.D. and Yukich, J.E. (2003). Weak laws of large numbers in geometric probability. Ann.

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[25] Reitzner, M., Spodarev, E. and Zaporozhets, D. (2012). Set reconstruction by Voronoi cells. Adv. inAppl. Probab. 44 938–953. MR3052844

[26] Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Probability and Its Applications.Berlin: Springer. MR2455326

[27] Schreiber, T. (2010). Limit theorems in stochastic geometry. In New Perspectives in Stochastic Geom-etry 111–144. Oxford: Oxford Univ. Press. MR2654677

[28] Schulte, M. (2012). A central limit theorem for the Poisson–Voronoi approximation. Adv. in Appl.Math. 49 285–306. MR3017961

[29] Tchoumatchenko, K. and Zuyev, S. (2001). Aggregate and fractal tessellations. Probab. Theory Re-lated Fields 121 198–218. MR1865485

[30] Yukich, J. (2013). Limit theorems in discrete stochastic geometry. In Stochastic Geometry, Spa-tial Statistics and Random Fields. Lecture Notes in Math. 2068 239–275. Heidelberg: Springer.MR3059650

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Bernoulli 22(4), 2016, 2401–2441DOI: 10.3150/15-BEJ733

Aggregation of autoregressive random fieldsand anisotropic long-range dependenceDONATA PUPLINSKAITE1 and DONATAS SURGAILIS2

1Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania.E-mail: donata.puplinskaite@mif.vu.lt2Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, LT-08663 Vilnius, Lithuania.E-mail: donatas.surgailis@mii.vu.lt

We introduce the notions of scaling transition and distributional long-range dependence for stationary ran-dom fields Y on Z

2 whose normalized partial sums on rectangles with sides growing at rates O(n) andO(nγ ) tend to an operator scaling random field Vγ on R

2, for any γ > 0. The scaling transition is char-acterized by the fact that there exists a unique γ0 > 0 such that the scaling limits Vγ are different and donot depend on γ for γ > γ0 and γ < γ0. The existence of scaling transition together with anisotropic andisotropic distributional long-range dependence properties is demonstrated for a class of α-stable (1 < α ≤ 2)

aggregated nearest-neighbor autoregressive random fields on Z2 with a scalar random coefficient A having

a regularly varying probability density near the “unit root” A = 1.

Keywords: α-stable mixed moving average; autoregressive random field; contemporaneous aggregation;isotropic/anisotropic long-range dependence; lattice Green function; operator scaling random field; scalingtransition

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Bernoulli 22(4), 2016, 2442–2485DOI: 10.3150/15-BEJ734

Asymptotic development for the CLT in totalvariation distanceVLAD BALLY1 and LUCIA CARAMELLINO2

1Université Paris-Est, LAMA (UMR CNRS, UPEMLV, UPEC), MathRisk INRIA, F-77454 Marne-la-Vallée,France. E-mail: bally@univ-mlv.fr2Dipartimento di Matematica, Università di Roma – Tor Vergata, Via della Ricerca Scientifica 1, I-00133Roma, Italy. E-mail: caramell@mat.uniroma2.it

The aim of this paper is to study the asymptotic expansion in total variation in the central limit theorem whenthe law of the basic random variable is locally lower-bounded by the Lebesgue measure (or equivalently,has an absolutely continuous component): we develop the error in powers of n−1/2 and give an explicitformula for the approximating measure.

Keywords: abstract Malliavin calculus; integration by parts; regularizing functions; total variation distance

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Bernoulli 22(4), 2016, 2486–2520DOI: 10.3150/15-BEJ735

Pathwise stochastic integrals for modelfree financeNICOLAS PERKOWSKI and DAVID J. PRÖMEL1CEREMADE & CNRS UMR 7534, Université Paris-Dauphine, France.E-mail: perkowski@ceremade.dauphine.fr2Humboldt-Universität zu Berlin, Institut für Mathematik, Germany. E-mail: proemel@math.hu-berlin.de

We present two different approaches to stochastic integration in frictionless model free financial mathemat-ics. The first one is in the spirit of Itô’s integral and based on a certain topology which is induced by theouter measure corresponding to the minimal superhedging price. The second one is based on the controlledrough path integral. We prove that every “typical price path” has a naturally associated Itô rough path, andjustify the application of the controlled rough path integral in finance by showing that it is the limit ofnon-anticipating Riemann sums, a new result in itself. Compared to the first approach, rough paths havethe disadvantage of severely restricting the space of integrands, but the advantage of being a Banach spacetheory.

Both approaches are based entirely on financial arguments and do not require any probabilistic structure.

Keywords: Föllmer integration; model uncertainty; rough path; stochastic integration; Vovk’s outermeasure

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Bernoulli 22(4), 2016, 2521–2547DOI: 10.3150/15-BEJ736

Methods for improving estimators oftruncated circular parametersKANIKA* and SOMESH KUMAR**

Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal,India. E-mail: *10ma90r04@iitkgp.ac.in; **smsh@maths.iitkgp.ernet.in

In decision theoretic estimation of parameters in Euclidean space Rp , the action space is chosen to be the

convex closure of the estimand space. In this paper, the concept has been extended to the estimation ofcircular parameters of distributions having support as a circle, torus or cylinder. As directional distributionsare of curved nature, existing methods for distributions with parameters taking values in R

p are not immedi-ately applicable here. A circle is the simplest one-dimensional Riemannian manifold. We employ conceptsof convexity, projection, etc., on manifolds to develop sufficient conditions for inadmissibility of estimatorsfor circular parameters. Further invariance under a compact group of transformations is introduced in theestimation problem and a complete class theorem for equivariant estimators is derived. This extends theresults of Moors [J. Amer. Statist. Assoc. 76 (1981) 910–915] on R

p to circles. The findings are of specialinterest to the case when a circular parameter is truncated. The results are implemented to a wide range ofdirectional distributions to obtain improved estimators of circular parameters.

Keywords: admissibility; convexity; directional data; invariance; projection; truncated estimation problem

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Bernoulli 22(4), 2016, 2548–2578DOI: 10.3150/15-BEJ737

An α-stable limit theorem under sublinearexpectationERHAN BAYRAKTAR* and ALEXANDER MUNK**

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA.E-mail: *erhan@umich.edu; **amunk@umich.edu

For α ∈ (1,2), we present a generalized central limit theorem for α-stable random variables under sublinearexpectation. The foundation of our proof is an interior regularity estimate for partial integro-differentialequations (PIDEs). A classical generalized central limit theorem is recovered as a special case, provided amild but natural additional condition holds. Our approach contrasts with previous arguments for the resultin the linear setting which have typically relied upon tools that are non-existent in the sublinear framework,for example, characteristic functions.

Keywords: generalized central limit theorem; partial-integro differential equations; stable distribution;sublinear expectation

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Bernoulli 22(4), 2016, 2579–2608DOI: 10.3150/15-BEJ738

Limit theorems for multifractal products ofgeometric stationary processesDENIS DENISOV1 and NIKOLAI LEONENKO2

1School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK.E-mail: denis.denisov@manchester.ac.uk; url: www.maths.manchester.ac.uk/~denisov/2School of Mathematics, Cardiff University, Senghennydd Road Cardiff CF24 4AG, UK.E-mail: LeonenkoN@cardiff.ac.uk

We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite andinfinite superpositions. We present the general conditions for the Lq convergence of cumulative processes tothe limiting processes and investigate their qth order moments and Rényi functions, which are non-linear,hence displaying the multifractality of the processes as constructed. We also establish the correspondingscenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normaltempered stable scenarios.

Keywords: geometric Gaussian process; geometric Ornstein–Uhlenbeck processes; Lévy processes;log-gamma scenario; log-normal scenario; log-normal tempered stable scenario; long-range dependence;log-variance gamma scenario; multifractal products; multifractal scenarios; Rényi function; scaling ofmoments; short-range dependence; stationary processes; superpositions

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Bernoulli 22(4), 2016, 2609–2614

Author Index

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