Summary
In this paper we consider the sequences of stochastic processes which converge weakly asn→∞ to Brownian local time. These processes are generated by a recurrent random walk with finite variance. The main result is the following: it is possible to redefine a random walk in such a way that for a wide class of processes the normalized differences between them and Brownian local time converge in distribution to some stochastic process. We also prove that such differences with probability one have the logarithmic upper bound. It is so called “Strong invariance principles for local times”.
References
Aleskeviciene, A.K.: On asymptotic distribution of local times of a recurrent random walk. In: Abstracts of Communications of IV USSR-Japan Symposium on Probability Theory and Mathematical Statistics vol.1, pp. 97–98. Tbilisi: Metsniereba 1982
Borodin, A.N.: An asymptotic behaviour of local times of a recurrent random walk with finite variance. Theory Probab. Appl.26, 769–783 (1981)
Borodin, A.N.: On distribution of integral type functionals of Brownian motion. Zapiski Nauchnych Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V.A.Steklova AN SSSR119, 19–38 (1982)
Borodin, A.N.: On the character of convergence to Brownian local time. Dokl. USSR Academy of Sciences269, 784–788 (1983)
Borodin, A.N.: On distribution of random walk local time. LOMI Preprint E-4-84. Leningrad 1984
Borodin, A.N.: On the character of convergence to Brownian local time I. Prob. Th. Rel. Fields72, 231–250 (1986)
Csáki, E., Révész, P.: Strong Invariance for local times. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 263–278 (1983)
Dobrushin, R.L.: Two limit theorems for simplest random walk on a line. Usp. Mat. Nauk10, 139–146 (1955)
Dobrushin, R.L.: The continuity condition for sample martingale functions. Theory Probab. Appl.3, 97–98 (1958)
Doob, J.L.: Stochastic processes, New York: Wiley 1953
Ito, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965
Kesten, H.: An iterated logarithm law for the local time. Duke Math. J.32, 447–456 (1965)
Knight, F.B.: Random walks and a sojourn density process of Brownian motion. Trans. Am. Math. Soc.109, 56–86 (1963)
McKean, H.P.: Stochastic integrals. New York-London: Academic Press 1969
Perkins, E.: Weak invariance principles for local time. Z. Wahrscheinlichkeitstheor. Verw. Geb.60,437–451 (1982)
Ray, D.B.: Sojourn times of a diffusion process. Ill. J. Math.7, 615–630 (1963)
Révész, P.: Local time and invariance. Lecture Notes in Math.861. Berlin-Heidelberg-New York: Springer 1981
Révész, P: A strong invariance principle of the local time of R.V.'s with continuous distribution. Stud. Sci. Math. Hung.16, 219–228 (1981)
Skorokhod A.V., Slobodenyuk, N.P.: Limit theorems for random walks. Kiev: Naukova Dumka 1970
Trotter, H.F.: A property of Brownian motion paths. Ill. J. Math.2, 425–433 (1958)
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Borodin, A.N. On the character of convergence to Brownian local time. II. Probab. Th. Rel. Fields 72, 251–277 (1986). https://doi.org/10.1007/BF00699106
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DOI: https://doi.org/10.1007/BF00699106
Keywords
- Stochastic Process
- Random Walk
- Probability Theory
- Local Time
- Mathematical Biology