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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.,
SOMMERFELD, M., WEICKERT, J. and WOLLNIK, C.
2113
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
PUPLINSKAITĖ, 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
RO S S A . M A L L E R
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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140 103–127. MR2357672
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Bernoulli 19 115–136. MR3019488
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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
C É L I N E L AC AU X 1,2,3 and G E N NA DY S A M O RO D N I T S K Y 4
1 Université de Lorraine, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506,
France. E-mail: [email protected]
2 CNRS, Institut Élie Cartan de Lorraine, UMR 7502, Vandœuvre-lès-Nancy, F-54506, France
3 Inria, BIGS, Villers-lès-Nancy, F-54600, France
4 School of Operations Research and Information Engineering and Department of Statistical Science Cornell 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 delicate 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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Bernoulli 22(4), 2016, 2001–2028
DOI: 10.3150/15-BEJ718
Quantifying repulsiveness of determinantal
point processes
C H R I S TO P H E A N G E NA P O L É O N B I S C I O 1 and F R É D É R I C L AVA N C I E R 2
1 Laboratoire de Mathématiques Jean Leray – BP 92208 – 2, Rue de la Houssinière – F-44322 Nantes
Cedex 03 – France. E-mail: [email protected]
2 Inria, 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 point process, 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 > 0 we
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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Bernoulli 22(4), 2016, 2029–2079
DOI: 10.3150/15-BEJ719
Cramér type moderate deviation theorems for
self-normalized processes
Q I - M A N S H AO 1 and W E N - X I N Z H O U 2,3
1 Department of Statistics, The Chinese University of Hong Kong, Shatin, NT, Hong Kong.
E-mail: [email protected]
2 Department of Operations Research and Financial Engineering, Princeton University, Princeton,
NJ 08544, USA. E-mail: [email protected]
3 School 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 approximation 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 deviation 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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Probab. 28 511–535. MR1756015
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Bernoulli 22(4), 2016, 2080–2100
DOI: 10.3150/15-BEJ720
Consistency of Bayes factor for nonnested
model selection when the model
dimension grows
M I N WA N G 1 and Y U Z O M A RU YA M A 2
1 Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931, USA.
E-mail: [email protected]
2 Center 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 regression 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 consistent when two competing models are of order O(nτ ) for τ < 1 and for τ = 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
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© 2016 ISI/BS
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Bernoulli 22(4), 2016, 2101–2112
DOI: 10.3150/15-BEJ721
A note on a local limit theorem for Wiener
space valued random variables
A L B E RTO L A N C O N E L L I 1 and AU R E L I . S TA N 2
1 Dipartimento di Matematica, Universitá degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari,
Italia. E-mail: [email protected]
2 Department 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
n . We close the paper comparing our
√
assumption the desired L1 -convergence of the density of X1 +···+X
n
result with certain Berry–Esseen bounds for multidimensional central limit theorems.
Keywords: abstract Wiener space; local limit theorem; Ornstein–Uhlenbeck semigroup; Wick product
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Bernoulli 22(4), 2016, 2113–2142
DOI: 10.3150/15-BEJ722
The circular SiZer, inferred persistence of
shape parameters and application to early
stem cell differentiation
S T E P H A N H U C K E M A N N 1,* , K WA N G - R A E K I M 2,** , A X E L M U N K 3,† ,
F L O R I A N R E H F E L D T 4,‡ , M A X S O M M E R F E L D 1,§ ,
J OAC H I M W E I C K E RT 5,¶ and C A R I NA WO L L N I K 4,
1 Felix Bernstein Institute for Mathematical Statistics in the Biosciences, University of Göttingen.
E-mail: * [email protected]; § [email protected]
2 School of Mathematical Sciences, University of Nottingham. E-mail: ** [email protected]
3 Max Planck Institute for Biophysical Chemistry, Göttingen and Felix Bernstein Institute for Mathematical
Statistics in the Biosciences, University of Göttingen. E-mail: † [email protected]
4 3rd Institute of Physics – Biophysics, University of Göttingen.
E-mail: ‡ [email protected]; [email protected]
5 Faculty 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
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Bernoulli 22(4), 2016, 2143–2176
DOI: 10.3150/15-BEJ724
Connection times in large ad-hoc
mobile networks
H A N NA D Ö R I N G 1 , G A B R I E L FA R AU D 2 and WO L F G A N G K Ö N I G 3,4
1 Universität Osnabrück, Institut für Mathematik, Albrechtstr. 28a, 49076 Osnabrück, Germany.
E-mail: [email protected]
2 Laboratoire Modal’x, Université Paris 10 Nanterre-La Défense, 200 Av. de la République, 92000 Nanterre,
France. E-mail: [email protected]
3 Weierstrass Institute Berlin, Mohrenstr. 39, 10117 Berlin, Germany. E-mail: [email protected]
4 Technische 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
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Bernoulli 22(4), 2016, 2177–2208
DOI: 10.3150/15-BEJ725
Integral approximation by kernel smoothing
B E R NA R D D E LYO N 1 and F R A N Ç O I S P O RT I E R 2
1 Institut de recherches mathématiques de Rennes (IRMAR), Campus de Beaulieu, Université de Rennes 1,
35042 Rennes Cédex, France. E-mail: [email protected]
2 Institut de Statistique, Biostatistique et Sciences Actuarielles (ISBA), Université catholique de Louvain,
Belgique. E-mail: [email protected]
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,
n
ϕ(Xi )
P
1/2
−1
n
− ϕ(x) dx −→ 0,
n
f( Xi )
i=1
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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Bernoulli 22(4), 2016, 2209–2236
DOI: 10.3150/15-BEJ726
The genealogy of a solvable population
model under selection with dynamics related
to directed polymers
A S E R C O RT I N E S
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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Bernoulli 22(4), 2016, 2237–2300
DOI: 10.3150/15-BEJ728
A stochastic algorithm finding p-means
on the circle
M A R C A R NAU D O N 1 and L AU R E N T M I C L O 2
1 Institut 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]
2 Institut de Mathématiques de Toulouse, UMR 5219, Université Toulouse 3 and CNRS, 118, route de Narbonne, 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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Bernoulli 22(4), 2016, 2301–2324
DOI: 10.3150/15-BEJ729
The combinatorial structure of beta negative
binomial processes
C R E I G H TO N H E AU K U L A N I 1 and DA N I E L M . ROY 2
1 Department of Engineering, University of Cambridge, Trumpington Street, Cambridge, CB2 1PZ, United
Kingdom. E-mail: [email protected]
2 Department 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 representation 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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Bernoulli 22(4), 2016, 2325–2371
DOI: 10.3150/15-BEJ731
Distributional representations and dominance
of a Lévy process over its maximal jump
processes
B O R I S BU C H M A N N 1,* , Y U G UA N G FA N 2 and RO S S A . M A L L E R 1,**
1 Research School of Finance, Actuarial Studies & Statistics, Mathematical Sciences Institute, Australian
National University, Australia. E-mail: * [email protected]; ** [email protected]
2 School 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 quadratic variation process Vt and its maximal jump processes, are derived, and used to make “small time” (as t ↓ 0) asymptotic comparisons between them. The
representations 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, they
enable 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 necessary and sufficient conditions for Xt / sup0<s≤t Xs and Xt / sup0<s≤t |Xs | to converge in probability
to 1, or to ∞, as t ↓ 0, so that X is either comparable to, or dominates, its largest jump. The former situation
tends to occur when the singularity at 0 of the Lévy measure of X is fairly mild (its tail is slowly varying
at 0), while the latter situation is related to the relative stability or attraction to normality of X at 0 (a steeper
singularity at 0). An important component in the analyses is the way the largest positive and negative jumps
interact 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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Bernoulli 22(4), 2016, 2372–2400
DOI: 10.3150/15-BEJ732
Asymptotic theory for statistics of the
Poisson–Voronoi approximation
C H R I S TO P H T H Ä L E 1 and J . E . Y U K I C H 2
1 Faculty of Mathematics, Ruhr University Bochum, Bochum, Germany. E-mail: [email protected]
2 Department 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 approximation 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 approximation. 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
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MR3297770
Bernoulli 22(4), 2016, 2401–2441
DOI: 10.3150/15-BEJ733
Aggregation of autoregressive random fields
and anisotropic long-range dependence
D O NATA P U P L I N S K A I T Ė1 and D O NATA S S U R G A I L I S 2
1 Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania.
E-mail: [email protected]
2 Institute 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 random fields Y on Z2 whose normalized partial sums on rectangles with sides growing at rates O(n) and
O(nγ ) tend to an operator scaling random field Vγ on R2 , for any γ > 0. The scaling transition is characterized 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
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Bernoulli 22(4), 2016, 2442–2485
DOI: 10.3150/15-BEJ734
Asymptotic development for the CLT in total
variation distance
V L A D BA L LY 1 and L U C I A C A R A M E L L I N O 2
1 Université Paris-Est, LAMA (UMR CNRS, UPEMLV, UPEC), MathRisk INRIA, F-77454 Marne-la-Vallée,
France. E-mail: [email protected]
2 Dipartimento 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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Bernoulli 22(4), 2016, 2486–2520
DOI: 10.3150/15-BEJ735
Pathwise stochastic integrals for model
free finance
N I C O L A S P E R KOW S K I and DAV I D J . P RÖ M E L
1 CEREMADE & CNRS UMR 7534, Université Paris-Dauphine, France.
E-mail: [email protected]
2 Humboldt-Universität zu Berlin, Institut für Mathematik, Germany. E-mail: [email protected]
We present two different approaches to stochastic integration in frictionless model free financial mathematics. 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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Bernoulli 22(4), 2016, 2521–2547
DOI: 10.3150/15-BEJ736
Methods for improving estimators of
truncated circular parameters
K A N I K A * and S O M E S H K U M A R **
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 immediately 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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Bernoulli 22(4), 2016, 2548–2578
DOI: 10.3150/15-BEJ737
An α-stable limit theorem under sublinear
expectation
E R H A N BAY R A K TA R * and A L E X A N D E R M U N K **
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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Bernoulli 22(4), 2016, 2579–2608
DOI: 10.3150/15-BEJ738
Limit theorems for multifractal products of
geometric stationary processes
D E N I S D E N I S OV 1 and N I KO L A I L E O N E N KO 2
1 School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK.
E-mail: [email protected]; url: www.maths.manchester.ac.uk/~denisov/
2 School 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 longrange 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 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 of
moments; short-range dependence; stationary processes; superpositions
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Bernoulli 22(4), 2016, 2609–2614
Author Index
1350-7265
© 2016 ISI/BS