The kinetic energy of taut strings accompanying the trajectory of a Wiener process or a random walk in a band of growing width is considered. It is shown that under certain assumptions on the band width, the energy obeys the same strong law of large numbers as in the previously studied case of constant width.
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References
M. Grasmair, “The equivalence of the taut string algorithm and BV-regularization,” J. Math. Imaging Vis., 27, 59–66 (2007).
Z. Kabluchko and M. Lifshits, “Least energy approximation for processes with stationary increments,” J. Theoret. Probab., 30, No. 1, 268–296 (2017).
J. Komlos, P. Major, and G. Tusnady, “An approximation of partial sums of independent RV’-s and the sample DF. I,” Z. Wahrscheinlichkeitstheor. Verw. Gebiete, 32, 111–131 (1975).
J. Komlos, P. Major, and G. Tusnady, “An approximation of partial sums of independent RV’-s and the sample DF. II,”Z. Wahrscheinlichkeitstheor. Verw. Gebiete, 34, 34–58 (1976).
N. Kruglyak and E. Setterqvist, “Discrete taut strings and real interpolation,” J. Func. Anal., 270, 671–704 (2016).
N. Kruglyak and E. Setterqvist, “Invariant K-minimal sets in the discrete and continuous setting,” J. Fourier Anal. Appl., 23, 572–611 (2017).
M. Lifshits, Gaussian Random Functions, Dordrecht, Kluwer (1995).
M. Lifshits and E. Setterqvist, “Energy of taut string accompanying Wiener process,” Stoch. Process. Appl., 125, 401–427 (2015).
M. Lifshits and A. Siuniaev, “Energy of taut strings accompanying random walk,” Probab. Math. Statist., 41, No. 1, 9–23 (2021).
P. Major, “The approximation of partial sums of independent r.v.’s,” Z. Wahrscheinlichkeitstheor. Verw. Gebiete, 35, 213–220 (1976).
E. Schertzer, “Renewal structure of the Brownian taut string,” Stoch. Process. Appl., 128, No. 2, 487–504 (2018).
O. Scherzer et al., Variational Methods in Imaging, Ser.: Applied Mathematical Sciences, 167, New York: Springer (2009).
A. I. Sakhanenko, “Rate of convergence in the invariance principle for variables with exponential moments that are not identically distributed,” Trudy Inst. Mat. Sib. Otd. AN SSSR, 3, 4–49 (1984).
A. Yu. Zaitsev, “The accuracy of strong Gaussian approximation for sums of independent random vectors,” Russian Math. Surveys, 68, No. 4, 721–761 (2013).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 495, 2020, pp. 64–86.
Translated by I. Ponomarenko.
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Blinova, D.I., Lifshits, M.A. Energy of Taut Strings Accompanying a Wiener Process and Random Walk in a Band of Variable Width. J Math Sci 268, 573–588 (2022). https://doi.org/10.1007/s10958-022-06228-6
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DOI: https://doi.org/10.1007/s10958-022-06228-6