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Linear rigidity of stationary stochastic processes

  • ALEXANDER I. BUFETOV (a1) (a2) (a3) (a4), YOANN DABROWSKI (a5) and YANQI QIU (a1)


We consider stationary stochastic processes $\{X_{n}:n\in \mathbb{Z}\}$ such that $X_{0}$ lies in the closed linear span of $\{X_{n}:n\neq 0\}$ ; following Ghosh and Peres, we call such processes linearly rigid. Using a criterion of Kolmogorov, we show that it suffices, for a stationary stochastic process to be linearly rigid, that the spectral density vanishes at zero and belongs to the Zygmund class $\unicode[STIX]{x1D6EC}_{\ast }(1)$ . We next give a sufficient condition for stationary determinantal point processes on $\mathbb{Z}$ and on $\mathbb{R}$ to be linearly rigid. Finally, we show that the determinantal point process on $\mathbb{R}^{2}$ induced by a tensor square of Dyson sine kernels is not linearly rigid.



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[1] Bufetov, A. I.. Rigidity of determinantal point processes with the Airy, the Bessel and the Gamma kernel. Bull. Math. Sci. 6(1) (2016), 163172.
[2] Costin, O. and Lebowitz, J. L.. Gaussian fluctuation in random matrices. Phys. Rev. Lett. 75(1) (1995), 6972.
[3] Daley, D. J. and Vere-Jones, D.. An Introduction to the Theory of Point Processes. Vol. I, Elementary Theory and Methods (Probability and its Applications) . Springer, New York, Berlin, Paris, 2003.
[4] Fülöp, V. and Móricz, F.. Absolutely convergent multiple Fourier series and multiplicative Zygmund classes of functions. Analysis 28(3) (2008), 345354.
[5] Ghosh, S.. Determinantal processes and completeness of random exponentials: the critical case. Probab. Theory Related Fields 163(3–4) (2015), 643665.
[6] Ghosh, S. and Peres, Y.. Rigidity and tolerance in point processes: Gaussian zeros and Ginibre eigenvalues. Duke Math. J., to appear. Preprint, 2012, arXiv:1211.3506.
[7] Holroyd, A. E. and Soo, T.. Insertion and deletion tolerance of point processes. Electron. J. Probab. 18(74) (2013), 24 pp.
[8] Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B.. Determinantal processes and independence. Probab. Surv. 3 (2006), 206229.
[9] Kolmogoroff, A. N.. Interpolation und Extrapolation von stationaeren zufaelligen Folgen. Izv. Math. (Izv. Akad. Nauk SSSR Ser. Mat.) 5(1) (1941), 314.
[10] Kolmogorov, A. N.. Stationary sequences in Hilbert space. Vestnik MGU (Bull. Univ. Moscou) 2(6) (1941), 140.
[11] Lyons, R.. Determinantal probability measures. Publ. Math. Inst. Hautes Études Sci. (98) (2003), 167212.
[12] Lyons, R. and Steif, J. E.. Stationary determinantal processes: phase multiplicity, Bernoullicity, entropy, and domination. Duke Math. J. 120(3) (2003), 515575.
[13] Macchi, O.. The coincidence approach to stochastic point processes. Adv. Appl. Probab. 7 (1975), 83122.
[14] Móricz, F.. Absolutely convergent Fourier series and function classes. J. Math. Anal. Appl. 324(2) (2006), 11681177.
[15] Nikolski, N. K.. Operators, Functions, and Systems: An Easy Reading. Vol. 1: Hardy, Hankel, and Toeplitz (Mathematical Surveys and Monographs, 92) . American Mathematical Society, Providence, RI, 2002, translated from the French by Andreas Hartmann.
[16] Soshnikov, A.. Determinantal random point fields. Uspekhi Mat. Nauk 55(5(335)) (2000), 107160.
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Ergodic Theory and Dynamical Systems
  • ISSN: 0143-3857
  • EISSN: 1469-4417
  • URL: /core/journals/ergodic-theory-and-dynamical-systems
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