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On the character of convergence to Brownian local time. II. (English) Zbl 0572.60079
[For part I see the foregoing review, Zbl 0572.60078.]
In this paper we consider the sequences of stochastic processes which converge weakly as $$n\to \infty$$ 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 the so called ”strong invariance principles for local times”.

##### MSC:
 60J65 Brownian motion 60J55 Local time and additive functionals
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##### References:
 [1] 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 [2] Borodin, A.N.: An asymptotic behaviour of local times of a recurrent random walk with finite variance. Theory Probab. Appl.26, 769-783 (1981) · Zbl 0474.60056 [3] 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) · Zbl 0491.60082 [4] Borodin, A.N.: On the character of convergence to Brownian local time. Dokl. USSR Academy of Sciences269, 784-788 (1983) · Zbl 0534.60032 [5] Borodin, A.N.: On distribution of random walk local time. LOMI Preprint E-4-84. Leningrad 1984 [6] Borodin, A.N.: On the character of convergence to Brownian local time I. Prob. Th. Rel. Fields72, 231-250 (1986) · Zbl 0572.60078 [7] Csáki, E., Révész, P.: Strong Invariance for local times. Z. Wahrscheinlichkeitstheor. Verw. Geb.62, 263-278 (1983) · Zbl 0488.60045 [8] Dobrushin, R.L.: Two limit theorems for simplest random walk on a line. Usp. Mat. Nauk10, 139-146 (1955) [9] Dobrushin, R.L.: The continuity condition for sample martingale functions. Theory Probab. Appl.3, 97-98 (1958) · Zbl 0082.34203 [10] Doob, J.L.: Stochastic processes, New York: Wiley 1953 · Zbl 0053.26802 [11] Ito, K., McKean, H.P.: Diffusion processes and their sample paths. Berlin-Heidelberg-New York: Springer 1965 · Zbl 0127.09503 [12] Kesten, H.: An iterated logarithm law for the local time. Duke Math. J.32, 447-456 (1965) · Zbl 0132.12701 [13] Knight, F.B.: Random walks and a sojourn density process of Brownian motion. Trans. Am. Math. Soc.109, 56-86 (1963) · Zbl 0119.14604 [14] McKean, H.P.: Stochastic integrals. New York-London: Academic Press 1969 · Zbl 0191.46603 [15] Perkins, E.: Weak invariance principles for local time. Z. Wahrscheinlichkeitstheor. Verw. Geb.60,437-451 (1982) · Zbl 0465.60065 [16] Ray, D.B.: Sojourn times of a diffusion process. Ill. J. Math.7, 615-630 (1963) · Zbl 0118.13403 [17] Révész, P.: Local time and invariance. Lecture Notes in Math.861. Berlin-Heidelberg-New York: Springer 1981 · Zbl 0456.60029 [18] 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) · Zbl 0525.60041 [19] Skorokhod A.V., Slobodenyuk, N.P.: Limit theorems for random walks. Kiev: Naukova Dumka 1970 · Zbl 0202.47403 [20] Trotter, H.F.: A property of Brownian motion paths. Ill. J. Math.2, 425-433 (1958) · Zbl 0117.35502
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