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Official Journal of the Bernoulli Society for Mathematical Statistics And Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Volume Twenty Six Number Two May 2020 ISSN: 1350-7265 CONTENTS DAVIS, R.A., NIELSEN, M.S. and ROHDE, V. 799 Stochastic differential equations with a fractionally filtered delay: A semimartingale model for long-range dependent processes HO, N., NGUYEN, X. and RITOV, Y. 828 Robust estimation of mixing measures in finite mixture models CÉNAC, P., DE LOYNES, B., OFFRET, Y. and ROUSSELLE, A. 858 Recurrence of multidimensional persistent random walks. Fourier and series criteria FAN, J.Y., HAMZA, K., JAGERS, P. and KLEBANER, F. 893 Convergence of the age structure of general schemes of population processes HIRSCH, C. and MÖNCH, C. 927 Distances and large deviations in the spatial preferential attachment model BONNET, G. and CHENAVIER, N. 948 The maximal degree in a Poisson–Delaunay graph DÖRING, L. and WEISSMANN, P. 980 Stable processes conditioned to hit an interval continuously from the outside MUKHERJEE, S. 1016 Degeneracy in sparse ERGMs with functions of degrees as sufficient statistics CONTI, P.L., MARELLA, D., MECATTI, F. and ANDREIS, F. 1044 A unified principled framework for resampling based on pseudo-populations: Asymptotic theory MARIUCCI, E., RAY, K. and SZABÓ, B. 1070 A Bayesian nonparametric approach to log-concave density estimation ALETTI, G., CRIMALDI, I. and GHIGLIETTI, A. 1098 Interacting reinforced stochastic processes: Statistical inference based on the weighted empirical means GAO, C. 1139 Robust regression via mutivariate regression depth LIU, Y. and PAGÈS, G. 1171 Characterization of probability distribution convergence in Wasserstein distance by Lp-quantization error function (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 Six Number Two May 2020 ISSN: 1350-7265 CONTENTS (continued) SCHWEINBERGER, M. 1205 Consistent structure estimation of exponential-family random graph models with block structure JIANG, B., CHEN, Z. and LENG, C. 1234 Dynamic linear discriminant analysis in high dimensional space GANTERT, N., HEYDENREICH, M. and HIRSCHER, T. 1269 Strictly weak consensus in the uniform compass model on Z DOLERA, E. and FAVARO, S. 1294 Rates of convergence in de Finetti’s representation theorem, and Hausdorff moment problem TALAY, D. and TOMAŠEVIC,´ M. 1323 A new McKean–Vlasov stochastic interpretation of the parabolic–parabolic Keller–Segel model: The one-dimensional case TKACHOV, P. 1354 On stability of traveling wave solutions for integro-differential equations related to branching Markov processes PAVLYUKEVICH, I. and SHEVCHENKO, G. 1381 Stratonovich stochastic differential equation with irregular coefficients: Girsanov’s example revisited WANG, W., SU, Z. and XIAO, Y. 1410 The moduli of non-differentiability for Gaussian random fields with stationary increments CONFORTI, G. and RIPANI, L. 1431 Around the entropic Talagrand inequality KOLB,M.andSAVOV,M. 1453 A characterization of the finiteness of perpetual integrals of Lévy processes BAI, S. and TAQQU, M.S. 1473 Limit theorems for long-memory flows on Wiener chaos KALBASI, K. and MOUNTFORD, T. 1504 On the probability distribution of the local times of diagonally operator-self-similar Gaussian fields with stationary increments NAJAFI, A., MOTAHARI, S.A. and RABIEE, H.R. 1535 Reliable clustering of Bernoulli mixture models JAUCH, M., HOFF, P.D. and DUNSON, D.B. 1560 Random orthogonal matrices and the Cayley transform KUCHIBHOTLA, A.K. and PATRA, R.K. 1587 Efficient estimation in single index models through smoothing splines Volume 26 Number 2 May 2020 Pages 799–1618 ISI/BS Volume 26 Number 2 May 2020 ISSN 1350-7265 BERNOULLI Official Journal of the Bernoulli Society for Mathematical Statistics and Probability Aims and Scope BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. Bernoulli Society for Mathematical Statistics and Probability The Bernoulli Society was founded in 1973. It is an autonomous Association of the International Sta- tistical Institute, ISI. According to its statutes, the object of the Bernoulli Society is the advancement, through international contacts, of the sciences of probability (including the theory of stochastic pro- cesses) and mathematical statistics and of their applications to all those aspects of human endeavour which are directed towards the increase of natural knowledge and the welfare of mankind. Meetings: http://www.bernoulli-society.org/index.php/meetings The Society holds a World Congress every four years; more frequent meetings, coordinated by the So- ciety’s standing committees and often organised in collaboration with other organisations, are the Eu- ropean Meeting of Statisticians, the Conference on Stochastic Processes and their Applications, the CLAPEM meeting (Latin-American Congress on Probability and Mathematical Statistics), the Euro- pean Young Statisticians Meeting, and various meetings on special topics – in the physical sciences in particular. The Society, as an association of the ISI, also collaborates with other ISI associations in the organization of the biennial ISI World Statistics Congresses (formerly ISI Sessions). Executive Committee The Society is headed by an Executive Committee. As of February 2020 the Executive Committee con- sists of: President: Claudia Klüppelberg (Germany); President Elect: Adam Jakubowski (Poland); Past President: Susan Murphy (USA); Treasurer: Geoffrey Grimmett (UK); Scientific Secretary: Song Xi Chen (China); Membership Secretary: Sebastian Engelke (Switzerland); Publicity Secretary: Leonardo Rolla (Argentina); Publication Secretary: Herold Dehling (Germany); ISI Director: Ada van Krimpen (Netherlands). Further, the Society has a twelve member Council and a number of standing committees to carry out the tasks outlined above. Final authority is the general assembly of members of the Society, meeting at least biennially at the ISI World Statistics Congresses. The papers published in Bernoulli are indexed or abstracted in Current Index to Statistics, Math- ematical Reviews, Statistical Theory and Method Abstracts-Zentralblatt (STMA-Z), Thomson Sci- entific and Zentralblatt für Mathematik (also available on the MATH via STN database and Com- pact 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 ©2020 International Statistical Institute/Bernoulli Society All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the Publisher. In 2020 Bernoulli consists of 4 issues published in February, May, August and November. Bernoulli 26(2), 2020, 799–827 https://doi.org/10.3150/18-BEJ1086 Stochastic differential equations with a fractionally filtered delay: A semimartingale model for long-range dependent processes RICHARD A. DAVIS1, MIKKEL SLOT NIELSEN2,* and VICTOR ROHDE2,** 1Department of Statistics, Columbia University, 1255 Amsterdam Avenue, New York, NY 10027, USA. E-mail: [email protected] 2Department of Mathematics, Aarhus University, Ny Munkegade 118, 8000 Aarhus C, Denmark. E-mail: *[email protected]; **[email protected] In this paper, we introduce a model, the stochastic fractional delay differential equation (SFDDE), which is based on the linear stochastic delay differential equation and produces stationary processes with hyper- bolically decaying autocovariance functions. The model departs from the usual way of incorporating this type of long-range dependence into a short-memory model as it is obtained by applying a fractional filter to the drift term rather than to the noise term. The advantages of this approach are that the corresponding long-range dependent solutions are semimartingales and the local behavior of the sample paths is unaffected by the degree of long memory. We prove existence and uniqueness of solutions to the SFDDEs and study their spectral densities and autocovariance functions. Moreover, we define a subclass of SFDDEs which we study in detail and relate to the well-known fractionally integrated CARMA processes. Finally, we consider the task of simulating from the defining SFDDEs. Keywords: long-range dependence; moving average processes; semimartingales; stochastic differential equations References [1] Barndorff-Nielsen, O.E. and Basse-O’Connor, A. (2011). Quasi Ornstein–Uhlenbeck processes. Bernoulli 17 916–941. MR2817611 https://doi.org/10.3150/10-BEJ311 [2] Basse-O’Connor, A., Nielsen, M.S., Pedersen, J. and Rohde, V. (2019). Stochastic delay differen- tial equations and related autoregressive models. Stochastics. https://doi.org/10.1080/17442508.2019. 1635601 [3] Basse-O’Connor, A., Nielsen, M.S., Pedersen, J. and Rohde, V. (2019). Multivariate
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