Geodesic
Energy of taut strings accompanying Wiener process and random walk in a band of variable width
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 64-86

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In the article, kinetic energy of taut strings accompanying trajectory of a Wiener process or the one of 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. 
@article{ZNSL_2020_495_a3,
     author = {D. I. Blinova and M. A. Lifshits},
     title = {Energy of taut strings accompanying {Wiener} process and random walk in a band of variable width},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {64--86},
     publisher = {mathdoc},
     volume = {495},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a3/}
}
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D. I. Blinova; M. A. Lifshits. Energy of taut strings accompanying Wiener process and random walk in a band of variable width. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 29, Tome 495 (2020), pp. 64-86. http://geodesic.mathdoc.fr/item/ZNSL_2020_495_a3/