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Limit theorems for local and occupation times of random walks and Brownian motion on a spider. (English) Zbl 1442.60081
Summary: A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs, we establish limit theorems for \(n\)-step local and occupation times.
MSC:
60J65 Brownian motion
60J55 Local time and additive functionals
60J60 Diffusion processes
60F15 Strong limit theorems
60G50 Sums of independent random variables; random walks
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[1] Appuhamillage, TA; Bokil, V.; Thomann, E.; Waymire, E.; Wood, B., Occupation and local times for skew Brownian motion with applications to dispersion across an interface, Ann. Appl. Probab., 21, 183-214, (2011) · Zbl 1226.60113
[2] Barlow, M.T., Pitman J.W., Yor M.: Une extension multidimensionelle de la loi de l’arc sinus. Sém. Prob. XXIII, Lect.Notes in Math.,vol. 1372. Springer, Berlin, Heidelberg, New York, pp. 294-314 (1989) · Zbl 0738.60072
[3] Bass, RF; Griffin, PS, The most visited site of Brownian motion and simple random walk, Z. Wahrsch. Verwandte Geb., 70, 417-436, (1985) · Zbl 0554.60076
[4] Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae. Birkhäuser, Basel (1996) · Zbl 0859.60001
[5] Burdzy, K.; Chen, Z-Q, Local time flow related to skew Brownian motion, Ann. Probab., 29, 1693-1715, (2001) · Zbl 1037.60057
[6] Cherny, A.; Shiryaev, A.; Yor, M., Limit behavior of the “horizontal-vertical” random walk and some extension of the Donsker-Prokhorov invariance principle, Theory Probab. Appl., 47, 377-394, (2004) · Zbl 1034.60076
[7] Chung, KL; Erdős, P., On the application of the Borel-Cantelli lemma, Trans. Am. Math. Soc., 64, 179-186, (1952) · Zbl 0046.35203
[8] Csáki, E.; Csörgő, M.; Földes, A.; Révész, P., How big are the increments of the local time of a Wiener process?, Ann. Probab., 11, 593-608, (1983) · Zbl 0545.60074
[9] Csáki, E.; Csörgő, M.; Földes, A.; Révész, P., Strong approximation of additive functionals, J. Theory Probab., 5, 679-706, (1992) · Zbl 0762.60024
[10] Csáki, E.; Csörgő, M.; Földes, A.; Révész, P., Global Strassen-type theorems for iterated Brownian motions, Stoch. Process. Appl., 59, 321-341, (1995) · Zbl 0843.60072
[11] Csáki, E.; Csörgő, M.; Földes, A.; Révész, P., Some limit theorems for heights of random walks on a spider, J. Theor. Probab., 29, 1685-1709, (2016) · Zbl 1359.60036
[12] Csáki, E.; Földes, A.; Révész, P., Strassen theorems for a class of iterated processes, Trans. Am. Math. Soc., 349, 1153-1167, (1997) · Zbl 0867.60051
[13] Csáki, E.; Salminen, P., On additive functionals of diffusion processes, Stud. Sci. Math. Hung., 31, 47-62, (1996) · Zbl 0851.60076
[14] Csörgő, M.; Horváth, L., On best possible approximations of local time, Stat. Probab. Lett., 8, 301-306, (1989) · Zbl 0691.60067
[15] Dobrushin, R.L.: Two limit theorems for the simplest random walk on a line. Usp. Mat. Nauk (N.S.) 10(3(65)), 139-146 (1955). (in Russian)
[16] Evans, SN, Snakes and spiders: Brownian motion on \(\mathbb{R}\)-trees, Probab. Theory Relat. Fields, 117, 361-386, (2000) · Zbl 0959.60070
[17] Gairat, A.; Shcherbakov, V., Density of skew Brownian motion and its functionals with application in finance, Math. Finance, 27, 1069-1088, (2017) · Zbl 1411.91555
[18] Hajri, H.: Discrete approximations to solution flows of Tanaka’s SDE related to Walsh Brownian motion. Sém. Prob. XLIV, Lect. Notes in Math., vol. 2046. Springer, Berlin, Heidelberg, New York, pp. 167-190 (2012) · Zbl 1261.60054
[19] Harrison, J.; Shepp, L., On skew Brownian motion, Ann. Probab., 9, 309-313, (1981) · Zbl 0462.60076
[20] Hu, Y.; Pierre-Loti-Viaud, D.; Shi, Z., Laws of the iterated logarithm for iterated Wiener processes, J. Theor. Probab., 8, 303-319, (1995) · Zbl 0816.60027
[21] Itô, K., McKean, H.: Diffusion and Their Sample Paths, 2nd edn. Springer, Berlin (1974) · Zbl 0285.60063
[22] Kesten, H., An iterated logarithm law for the local time, Duke Math. J., 32, 447-456, (1965) · Zbl 0132.12701
[23] Komlós, M.; Major, P.; Tusnády, G., An approximation of partial sums of independent r.v’.s and sample df, I. Z. Wahrsch. Verwandte Geb., 32, 111-131, (1975) · Zbl 0308.60029
[24] Lamperti, J., An occupation time theorem for a class of stochastic processes, Trans. Am. Math. Soc., 88, 380-387, (1958) · Zbl 0228.60046
[25] Lejay, A., On the constructions of the skew Brownian motion, Probab. Surv., 3, 413-466, (2006) · Zbl 1189.60145
[26] Lejay, A.: Estimation of the bias parameter of the Skew Random Walk and application to the Skew Brownian Motion. Stat. Inference Stoch. Process (2017). doi:10.1007/s11203-017-9161-9
[27] Lyulko, YA, On the distribution of time spent by a Markov chain at different levels until achieving a fixed state, Theory Probab. Appl., 56, 140-149, (2012) · Zbl 1247.60108
[28] Petrov, V.V.: Sums of Independent Random Variables. Springer, Berlin (1975) · Zbl 0322.60042
[29] Révész, P.: Random Walk in Random and Non-random Environment, 3d edn. World Scientific, Singapore (2013) · Zbl 1283.60007
[30] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1991) · Zbl 0731.60002
[31] Spitzer, F.: Principles of Random Walk. Van Nostrand, Princeton (1964) · Zbl 0119.34304
[32] Walsh, JB, A diffusion with discontinuous local time, Astérisque, 52-53, 37-45, (1978)
[33] Watanabe, S.: Generalized arc-sine laws for one-dimensional diffusion processes and random walks. In: Proceedings of Symposia in Pure Mathematics, vol. 57. Stoch. Analysis, Cornell University (1993), Am. Math. Soc., pp. 157-172 (1995) · Zbl 0824.60080
[34] Yano, Y., On the joint law of the occupation times for a diffusion process on multiray, J. Theor. Probab., 30, 490-509, (2017) · Zbl 1374.60155
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