Official Journal of the Bernoulli Society for Mathematical and

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 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

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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

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Quantifying repulsiveness of determinantal point processes

CHRISTOPHE ANGE NAPOLÉON BISCIO1 and FRÉDÉRIC LAVANCIER2 1Laboratoire de Mathématiques Jean Leray – BP 92208 – 2, Rue de la Houssinière – F-44322 Nantes Cedex 03 – France. E-mail: [email protected] 2Inria, Centre Rennes Bretagne Atlantique, France. E-mail: [email protected]

Determinantal point processes (DPPs) have recently proved to be a useful class of models in several areas of statistics, including spatial statistics, statistical learning and telecommunications networks. They are models for repulsive (or regular, or inhibitive) point processes, in the sense that nearby points of the process tend to repel each other. We consider two ways to quantify the repulsiveness of a , both based on 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>0we investigate repulsiveness in the subclass of R-dependent stationary DPPs, that is, stationary DPPs with R-compactly supported kernels. Finally, in both the general case and the R-dependent case, we present some new parametric families of stationary DPPs that can cover a large range of DPPs, from the stationary Poisson 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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Cramér type moderate deviation theorems for self-normalized processes

QI-MAN SHAO1 and WEN-XIN ZHOU2,3 1Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong. E-mail: [email protected] 2Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ 08544, USA. E-mail: [email protected] 3School 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 this paper, 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 nonlinear statistics. In particular, a sharp moderate deviation theorem under optimal moment conditions is established for Studentized U-statistics. Keywords: moderate deviation; nonlinear statistics; relative error; self-normalized processes; Studentized statistics; U-statistics

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Consistency of Bayes factor for nonnested model selection when the model dimension grows

MIN WANG1 and YUZO MARUYAMA2 1Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA. E-mail: [email protected] 2Center for Spatial Information Science, University of Tokyo, Bunkyo-ku, Tokyo, 113-0033, Japan. E-mail: [email protected]

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 prior and put a special hyper-prior for g, which results in a closed-form expression of Bayes factor for nested linear 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<1andforτ = 1 is almost consistent except a small inconsistency region around the null hypothesis. In this paper, we study Bayes factor consistency for nonnested linear models with a growing number of parameters. Some of the proposed results generalize the ones of the Bayes factor for the case of nested linear models. Specifically, we compare the asymptotic behaviors between the proposed Bayes factor and the intrinsic Bayes factor in the literature.

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

References

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A note on a local limit theorem for Wiener space valued random variables

ALBERTO LANCONELLI1 and AUREL I. STAN2 1Dipartimento di Matematica, Universitá degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italia. E-mail: [email protected] 2Department of Mathematics, Ohio State University at Marion, 1465 Mount Vernon Avenue, Marion, OH 43302, USA. E-mail: [email protected]

We prove a local limit theorem, that is, a central limit theorem for densities, for a sequence of independent and identically distributed random variables taking values on an abstract Wiener space; the common law of those random variables is assumed to be absolutely continuous with respect to the reference Gaussian measure. We begin by showing that the key roles of scaling operator and convolution product in this infinite dimensional 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 variables for the local limit theorem to hold true. We then reverse the implication and prove under an additional +···+ assumption the desired L1-convergence of the density of X1 √ Xn . We close the paper comparing our n 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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The circular SiZer, inferred persistence of shape parameters and application to early stem cell differentiation

STEPHAN 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: *[email protected]; §[email protected] 2School of Mathematical Sciences, University of Nottingham. E-mail: **[email protected] 3Max Planck Institute for Biophysical Chemistry, Göttingen and Felix Bernstein Institute for Mathematical Statistics in the Biosciences, University of Göttingen. E-mail: †[email protected] 43rd Institute of Physics – Biophysics, University of Göttingen. E-mail: ‡[email protected]; [email protected] 5Faculty of Mathematics and Computer Science, Saarland University. E-mail: ¶[email protected]

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 circular data. It turns out that only the wrapped Gaussian kernel gives a symmetric, strongly Lipschitz semi-group satisfying “circular” causality, that is, not introducing possibly artificial modes with increasing levels of smoothing. 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-induced differentiation in adult human stem cells from their actin-myosin filament structure. As a consequence, the circular SiZer based on the wrapped Gaussian kernel (WiZer) allows the verification at a controlled error level of the observation reported by Zemel et al. (Nat. Phys. 6 (2010) 468–473): Within early stem cell differentiation, 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

References

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Connection times in large ad-hoc mobile networks

HANNA DÖRING1, GABRIEL FARAUD2 and WOLFGANG KÖNIG3,4 1Universität Osnabrück, Institut für Mathematik, Albrechtstr. 28a, 49076 Osnabrück, Germany. E-mail: [email protected] 2Laboratoire Modal’x, Université Paris 10 Nanterre-La Défense, 200 Av. de la République, 92000 Nanterre, France. E-mail: [email protected] 3Weierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germany. E-mail: [email protected] 4Technische 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 consider a large number of participants of the system moving randomly, independently and identically distributed in a large domain, with a space-dependent population density of finite, positive order and with a fixed time horizon. Messages are instantly transmitted according to a relay principle, that is, they are iteratively forwarded from participant to participant over distances smaller than the communication radius until they reach 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 two are connected with each other. In the above thermodynamic limit, we find that the connectivity induced by the system can be described in terms of the counterplay of a local, random and a global, deterministic mechanism, 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 probability of the event that the portion of the connection time is less than the expectation.

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

References

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Integral approximation by kernel smoothing

BERNARD DELYON1 and FRANÇOIS PORTIER2 1Institut de recherches mathématiques de Rennes (IRMAR), Campus de Beaulieu, Université de Rennes 1, 35042 Rennes Cédex, France. E-mail: [email protected] 2Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain, Belgique. E-mail: [email protected]

d Let (X1,...,Xn) be an i.i.d. sequence of random variables in R , d ≥ 1. We show that, for any function ϕ : Rd → R, under regularity conditions,   n  − ϕ(X ) P n1/2 n 1 i − ϕ(x)dx −→ 0, f(X ) i=1 i  where f is the classical kernel estimator of the density of X1. This result is striking because it speeds up traditional rates, in root n, derived from the central limit theorem when f= f . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions ϕ and f , the dimension d and the bandwidth of the kernel estimator f. Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of ϕ. As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.

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

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The genealogy of a solvable population model under selection with dynamics related to directed polymers

ASER CORTINES Université Paris Diderot, Mathématiques, case 7012, F-75 205 Paris Cedex 13, France. E-mail: [email protected]

We consider a stochastic model describing a constant size N population that may be seen as a directed polymer in random medium with N sites in the transverse direction. The population dynamics is governed by a noisy traveling wave equation describing the evolution of the individual fitnesses. We show that under suitable 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 show that the average coalescence times scales like log N and that the limit genealogical trees are governed by the Bolthausen–Sznitman coalescent, which validates the predictions by Brunet, Derrida, Mueller and Munier for this class of models. We also study the extended Wright–Fisher model, and show that, under certain conditions on ηi, the limit may be Kingman’s coalescent, a coalescent with multiple collisions, or a coalescent with simultaneous multiple collisions.

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

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A stochastic algorithm finding p-means on the circle

MARC ARNAUDON1 and LAURENT MICLO2 1Institut de Mathématique de Bordeaux, UMR 5251, Université de Bordeaux and CNRS, 351, Cours de la Libération, F-33405 TALENCE Cedex, France. E-mail: [email protected] 2Institut de Mathématiques de Toulouse, UMR 5219, Université Toulouse 3 and CNRS, 118, route de Nar- bonne, 31062 Toulouse Cedex 9, France. E-mail: [email protected]

A stochastic algorithm is proposed, finding some elements from the set of intrinsic p-mean(s) associated to a probability measure ν on a compact Riemannian manifold and to p ∈[1, ∞). It is fed sequentially with independent random variables (Yn)n∈N distributed according to ν, which is often the only available knowledge of ν. Furthermore, the algorithm is easy to implement, because it evolves like a Brownian motion between the random times when it jumps in direction of one of the Yn, n ∈ N. Its principle is based on simulated annealing and homogenization, so that temperature and approximations schemes must be tuned up (plus a regularizing scheme if ν does not admit a Hölderian density). The analysis of the convergence is restricted to the case where the state space is a circle. In its principle, the proof relies on the investigation of the evolution of a time-inhomogeneous L2 functional and on the corresponding spectral gap estimates due to Holley, Kusuoka and Stroock. But it requires new estimates on the discrepancies between the unknown instantaneous 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 small temperature; stochastic algorithms

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The combinatorial structure of beta negative binomial processes

CREIGHTON HEAUKULANI1 and DANIEL M. ROY2 1Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United Kingdom. E-mail: [email protected] 2Department of Statistical Sciences, University of Toronto, 100 St. George Street, Toronto, ON, M5S 3G3, Canada. E-mail: [email protected]

We characterize the combinatorial structure of conditionally-i.i.d. sequences of negative binomial processes with a common beta process base measure. In Bayesian nonparametric applications, such processes have served as models for latent multisets of features underlying data. Analogously, random subsets arise from conditionally-i.i.d. sequences of Bernoulli processes with a common beta process base measure, in which case the combinatorial structure is described by the Indian buffet process. Our results give a count analogue of the Indian buffet process, which we call a negative binomial Indian buffet process. As an intermediate step 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 use a NB-IBP representation in a Markov Chain Monte Carlo algorithm targeting a posterior distribution.

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

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Distributional representations and dominance of a Lévy process over its maximal jump processes

BORIS BUCHMANN1,*, YUGUANG FAN2 and ROSS A. MALLER1,** 1Research School of Finance, Actuarial Studies & Statistics, Mathematical Sciences Institute, Australian National University, Australia. E-mail: *[email protected]; **[email protected] 2School of Mathematics & Statistics, University of Melbourne, ARC Centre of Excellence for Mathematics & Statistical Frontiers, Australia. E-mail: [email protected]

Distributional identities for a Lévy process Xt , its process Vt and its maximal jump pro- cesses, are derived, and used to make “small time” (as t ↓ 0) asymptotic comparisons between them. The representations are constructed using properties of the underlying of the jumps of X. Apart from providing insight into the connections between X, V , and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study “self-normalised” | | versions of Xt ,thatis,Xt after division by sup0

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

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Asymptotic theory for statistics of the Poisson–Voronoi approximation

CHRISTOPH THÄLE1 and J.E. YUKICH2 1Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany. E-mail: [email protected] 2Department of Mathematics, Lehigh University, Bethlehem, PA 18015, USA. E-mail: [email protected]

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. Statistics of interest include volume, surface area, Hausdorff measure, and the number of faces of lower-dimensional skeletons. 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 formula applicable in the setting of sums of stabilizing functionals.

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

References

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Aggregation of autoregressive random fields and anisotropic long-range dependence

DONATA PUPLINSKAITE˙ 1 and DONATAS SURGAILIS2 1Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania. E-mail: [email protected] 2Institute of Mathematics and Informatics, Vilnius University, Akademijos 4, LT-08663 Vilnius, Lithuania. E-mail: [email protected]

We introduce the notions of scaling transition and distributional long-range dependence for stationary ran- dom fields Y on Z2 whose normalized partial sums on rectangles with sides growing at rates O(n) and γ 2 O(n ) tend to an operator scaling random field Vγ on R ,foranyγ>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 do not depend on γ for γ>γ0 and γ<γ0. The existence of scaling transition together with anisotropic and isotropic 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; scaling transition

References

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Asymptotic development for the CLT in total variation distance

VLAD BALLY1 and LUCIA CARAMELLINO2 1Université Paris-Est, LAMA (UMR CNRS, UPEMLV, UPEC), MathRisk INRIA, F-77454 Marne-la-Vallée, France. E-mail: [email protected] 2Dipartimento di Matematica, Università di Roma – Tor Vergata, Via della Ricerca Scientifica 1, I-00133 Roma, Italy. E-mail: [email protected]

The aim of this paper is to study the asymptotic expansion in total variation in the central limit theorem when the 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 explicit formula for the approximating measure.

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

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Pathwise stochastic integrals for model free finance

NICOLAS PERKOWSKI and DAVID J. PRÖMEL 1CEREMADE & CNRS UMR 7534, Université Paris-Dauphine, France. E-mail: [email protected] 2Humboldt-Universität zu Berlin, Institut für Mathematik, Germany. E-mail: [email protected]

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 the outer measure corresponding to the minimal superhedging price. The second one is based on the controlled rough path integral. We prove that every “typical price path” has a naturally associated Itô rough path, and justify the application of the controlled rough path integral in finance by showing that it is the limit of non-anticipating Riemann sums, a new result in itself. Compared to the first approach, rough paths have the disadvantage of severely restricting the space of integrands, but the advantage of being a Banach space theory. 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 outer measure

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Methods for improving estimators of truncated circular parameters

KANIKA* and SOMESH KUMAR** Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, West Bengal, India. E-mail: *[email protected]; **[email protected]

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 of circular parameters of distributions having support as a circle, torus or cylinder. As directional distributions are of curved nature, existing methods for distributions with parameters taking values in Rp are not immedi- ately applicable here. A circle is the simplest one-dimensional Riemannian manifold. We employ concepts of convexity, projection, etc., on manifolds to develop sufficient conditions for inadmissibility of estimators for circular parameters. Further invariance under a compact group of transformations is introduced in the estimation problem and a complete class theorem for equivariant estimators is derived. This extends the results of Moors [J. Amer. Statist. Assoc. 76 (1981) 910–915] on Rp to circles. The findings are of special interest to the case when a circular parameter is truncated. The results are implemented to a wide range of directional distributions to obtain improved estimators of circular parameters.

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

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An α-stable limit theorem under sublinear expectation

ERHAN BAYRAKTAR* and ALEXANDER MUNK** Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA. E-mail: *[email protected]; **[email protected]

For α ∈ (1, 2), we present a generalized central limit theorem for α-stable random variables under sublinear expectation. The foundation of our proof is an interior regularity estimate for partial integro-differential equations (PIDEs). A classical generalized central limit theorem is recovered as a special case, provided a mild but natural additional condition holds. Our approach contrasts with previous arguments for the result in 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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Limit theorems for multifractal products of geometric stationary processes

DENIS DENISOV1 and NIKOLAI LEONENKO2 1School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK. E-mail: [email protected];url:www.maths.manchester.ac.uk/~denisov/ 2School of Mathematics, Cardiff University, Senghennydd Road Cardiff CF24 4AG, UK. E-mail: [email protected]

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 and infinite superpositions. We present the general conditions for the Lq convergence of cumulative processes to the 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 corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios.

Keywords: geometric ; 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 of moments; short-range dependence; stationary processes; superpositions

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