×

zbMATH — the first resource for mathematics

Processes iterated ad libitum. (English) Zbl 1348.60026
Summary: Consider the \(n\)-th iterated Brownian motion \(I^{(n)} = B_n \circ \cdots \circ B_1\). N. Curien and T. Konstantopoulos [J. Theor. Probab. 27, No. 2, 433–448 (2014; Zbl 1331.60159)] proved that for any distinct numbers \(t_i \neq 0\), \((I^{(n)}(t_1), \ldots, I^{(n)}(t_k))\) converges in distribution to a limit \(I[k]\) independent of the \(t_i\)’s, exchangeable, and gave some elements on the limit occupation measure of \(I^{(n)}\). Here, we prove under some conditions that the finite-dimensional distributions of \(n\)-times iterated two-sided stable processes converge, and the same holds the reflected Brownian motions. We give a description of the law of \(I[k]\) and of the finite-dimensional distributions of \(I^{(n)}\), as well as those of the iterated reflected Brownian motion iterated ad libitum.

MSC:
60F05 Central limit and other weak theorems
60G52 Stable stochastic processes
60J65 Brownian motion
60G09 Exchangeability for stochastic processes
60G57 Random measures
60J05 Discrete-time Markov processes on general state spaces
60E99 Distribution theory
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Applebaum, D., (Lévy Processes and Stochastic Calculus, Cambridge Studies in Advanced Mathematics, (2004), Cambridge University Press)
[2] Barnsley, M. F.; Demko, S., Iterated function systems and the global construction of fractals, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 399, 1817, 243-275, (1985) · Zbl 0588.28002
[3] Barnsley, M.; Demko, S.; Elton, J.; Geronimo, J., Invariant measures for Markov processes arising from iterated function systems with place-dependent probabilities, Ann. Inst. Henri Poincaré Probab. Stat., 24, 3, 367-394, (1988) · Zbl 0653.60057
[4] Bertoin, J., Iterated Brownian motion and stable(1/4) subordinator, Statist. Probab. Lett., 27, 2, 111-114, (1996) · Zbl 0854.60082
[5] Burdzy, K., Some path properties of iterated Brownian motion, (Seminar on Stochastic Processes, 1992, (1993), Springer), 67-87 · Zbl 0789.60060
[6] Burdzy, K.; Khoshnevisan, D., The level sets of iterated Brownian motion, (Séminaire de Probabilités XXIX, (1995), Springer), 231-236 · Zbl 0853.60061
[7] Curien, N.; Konstantopoulos, T., Iterating Brownian motions, ad libitum, J. Theoret. Probab., 27, 2, 433-448, (2014) · Zbl 1331.60159
[8] Eisenbaum, N.; Shi, Z., Uniform oscillations of the local time of iterated Brownian motion, Bernoulli, 5, 1, 49-65, (1999) · Zbl 0930.60056
[9] Falconer, K., Fractal geometry: mathematical foundations and applications, (2004), Wiley
[10] Feller, W., An introduction to probability theory and its applications. vol. II, (1971), John Wiley & Sons Inc. New York · Zbl 0219.60003
[11] Funaki, T., Probabilistic construction of the solution of some higher order parabolic differential equation, Proc. Japan Acad. Ser. A Math. Sci., 55, 5, 176-179, (1979) · Zbl 0433.35039
[12] Hutchinson, J. E., Fractals and self-similarity, Indiana Univ. Math. J., 30, 5, 713-747, (1981) · Zbl 0598.28011
[13] Khoshnevisan, D.; Lewis, T. M., Iterated Brownian motion and its intrinsic skeletal structure, (Seminar on Stochastic Analysis, Random Fields and Applications, (1999), Springer), 201-210 · Zbl 0943.60081
[14] Meyn, S.; Tweedie, R. L., Markov chains and stochastic stability, (2009), Cambridge University Press New York, NY, USA · Zbl 0925.60001
[15] Orsingher, E.; Beghin, L., Fractional diffusion equations and processes with randomly varying time, Ann. Probab., 206-249, (2009) · Zbl 1173.60027
[16] Pitt, L., Local times for Gaussian vector fields, Indiana Univ. Math. J., 27, 309-330, (1978) · Zbl 0382.60055
[17] Revuz, D.; Yor, M., (Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenchaften A Series of Comprehensive Studies in Mathematics, (1999), Springer)
[18] Turban, L., Iterated random walk, Europhys. Lett., 65, 5, 627, (2004)
[19] Xiao, Y., Local times and related properties of multidimensional iterated Brownian motion, J. Theoret. Probab., 11, 2, 383-408, (1998) · Zbl 0914.60063
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.