Geodesic
Approximation of a Wiener process local time by functionals of random walks
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 1, pp. 73-93 Cet article a éte moissonné depuis la source Math-Net.Ru

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A sequence of compound Poisson processes constructed from sums of identically distributed random variables that weakly converges to a Wiener process is considered. Certain functionals of these processes are shown to converge in distribution to the local time of a Wiener process.
Keywords: random process, limit theorem, local time. 
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I. A. Ibragimov; N. V. Smorodina; M. M. Faddeev. Approximation of a Wiener process local time by functionals of random walks. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 1, pp. 73-93. http://geodesic.mathdoc.fr/item/TVP_2021_66_1_a3/

[1] E. S. Boylan, “Local times for a class of Markoff processes”, Illinois J. Math., 8 (1964), 19–39 | DOI | MR | Zbl

[2] R. M. Blumenthal, R. K. Getoor, “Local times for Markov processes”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3 (1964), 50–74 | DOI | MR | Zbl

[3] A. N. Borodin, I. A. Ibragimov, “Limit theorems for functionals of random walks”, Proc. Steklov Inst. Math., 195 (1995), 1–259 | MR | Zbl

[4] S. M. Berman, “Gaussian processes with stationary increments: local times and sample function properties”, Ann. Math. Statist., 41:4 (1970), 1260–1272 | DOI | MR | Zbl

[5] S. M. Berman, “Local times and sample function properties of stationary Gaussian process”, Trans. Amer. Math. Soc., 137 (1969), 277–299 | DOI | MR | Zbl

[6] H. F. Trotter, “A property of Brownian motion paths”, Illinois J. Math., 2:3 (1958), 425–433 | DOI | MR | Zbl

[7] B. B. Bergery, P. Vallois, “Approximation via regularization of the local time of semimartingales and Brownian motion”, Stochastic Process. Appl., 118:11 (2008), 2058–2070 | DOI | MR | Zbl

[8] Kai Lai Chung, Zhongxin Zhao, From Brownian motion to Schrödinger's equation, Grundlehren Math. Wiss., 312, Springer-Verlag, Berlin, 1995, xii+287 pp. | DOI | MR | Zbl

[9] P. Billingsley, Convergence of probability measures, John Wiley Sons, Inc., New York–London–Sydney, 1968, xii+253 pp. | MR | MR | Zbl

[10] I. A. Ibragimov, N. V. Smorodina, M. M. Faddeev, “Ob odnom obobschenii ponyatiya lokalnogo vremeni”, Veroyatnost i statistika. 28, Zap. nauch. sem. POMI, 486, POMI, SPb., 2019, 148–157

[11] I. I. Gikhman, A. V. Skorokhod, Introduction to the theory of random processes, W. B. Saunders Co., Philadelphia, PA–London–Toronto, ON, 1969, xiii+516 pp. | MR | MR | Zbl

[12] M. Sh. Birman, M. Z. Solomjak, Spectral theory of self-adjoint operators in Hilbert space, Math. Appl. (Soviet Ser.), 5, D. Reidel Publ. Co., Dordrecht, 1987, xv+301 pp. | MR | Zbl

[13] A. M. Savchuk, A. A. Shkalikov, “Sturm–Liouville operators with singular potentials”, Math. Notes, 66:6 (1999), 741–753 | DOI | DOI | MR | Zbl

[14] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, H. Holden, Solvable models in quantum mechanics, Texts Monogr. Phys., Springer-Verlag, New York, 1988, xiv+452 pp. | DOI | MR | MR | Zbl

[15] I. M. Gelfand, G. E. Schilow, Verallgemeinerte Funktionen (Distributionen), v. I, Hochschulbücher für Math., 47, Funktionen und das Rechnen mit ihnen, VEB Deutscher Verlag der Wissenschaften, Berlin, 1960, 364 pp. | MR | MR | Zbl | Zbl

[16] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag New York, Inc., New York, 1966, xix+592 pp. | DOI | MR | MR | Zbl | Zbl

[17] M. Reed, B. Simon, Methods of modern mathematical physics, v. III, Scattering theory, Academic Press, Inc., New York–London, 1979, xv+463 pp. | MR | MR | Zbl | Zbl

[18] N. Dunford, J. T. Schwartz, Linear operators, v. I, Pure Appl. Math., 7, General theory, Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958, xiv+858 pp. | MR | MR | Zbl

[19] R. K. Getoor, “Another limit theorem for local time”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 34:1 (1976), 1–10 | DOI | MR | Zbl