×

zbMATH — the first resource for mathematics

Iterated Brownian motion and stable \((1/4)\) subordinator. (English) Zbl 0854.60082
Let \(B_t\), \(B^+_t\), \(B^-_t\), \(t \geq 0\), be three independent linear Brownian motions started from 0. Denote \(S_t = \sup \{B_s :0 \leq s \leq t\}\) and \(\sigma_t = \inf \{u : S_u > t\}\). Consider also the supremum processes \(S^+\), \(S^-\) of \(B^+\) and \(B^-\), respectively, denote \(\sigma^+\), \(\sigma^-\) the right-continuous inverses of \(B^+\) and \(B^-\), respectively. The author notices that the compound process \(\sigma \circ \sigma^+_t\) is a stable subordinator with index \(\alpha = 1/4\) and proposes a very easy method to investigate the path behaviour of iterated Brownian motion \(X_t\). Here the process \(X_t\), \(t \geq 0\), is given by expressions \(X_t = B^+ (B_t)\) if \(B_t \geq 0\) and \(X_t = B^-(- B_t)\) if \(B_t < 0\). Based on known properties of stable processes with \(\alpha \in (0,1)\) the LIL for \(X_t\) is of the form: \[ \limsup t^{-1/4} (\log |\log t |)^{- 3/4} X_t = 2^{5/4} 3^{-3/4}. \] Both as \(t \to 0\) and \(t \to \infty\) are proved. Also the integral tests to study the rate of grows for \(S^+ \circ S\) and \(X_t\) are proposed and modulas of continuity of the supremum of iterated Brownian motion are obtained.

MSC:
60J65 Brownian motion
60J99 Markov processes
60G17 Sample path properties
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bochner, S., Harmonic analysis and the theory of probability, (1955), Univ. California Press Berkeley, CA · Zbl 0068.11702
[2] Breiman, L.; Breiman, L., A delicate law of the iterated logarithm for non-decreasing stable processes, Ann. math. statist., Ann. math. statist., 41, 1126-1127, (1970), Correction id. · Zbl 0207.48601
[3] Burdzy, K., Some path properties of iterated Brownian motion, (), 67-87 · Zbl 0789.60060
[4] Burdzy, K.; Khoshnevisan, D., The level sets of iterated Brownian motion, Séminaire de probabilités XXIX, (), 231-236 · Zbl 0853.60061
[5] Csáki, E.; Csörgo&#x030B;, M.; Földes, A.; Révész, P., Brownian local time approximated by a Wiener sheet, Ann. probab., 17, 516-537, (1989) · Zbl 0674.60072
[6] Deheuvels, P.; Mason, D.M., A functional LIL approach to pointwise bahadur-kiefer theorems, (), 255-266 · Zbl 0844.60012
[7] Fristedt, B.E., The behavior of increasing stable processes for both small and large times, J. math. mech., 13, 849-856, (1964) · Zbl 0126.33403
[8] Fristedt, B.E., Sample functions of stochastic processes with stationary independent increments, (), 241-396 · Zbl 0189.50802
[9] Funaki, T., A probabilistic construction of the solution of some higher order parabolic differential equations, (), 176-179 · Zbl 0433.35039
[10] Hawkes, J., A lower Lipschitz condition for the stable subordinator, Z. wahrsch. verw. geb., 17, 23-32, (1971) · Zbl 0193.45002
[11] Hirsch, W.M., A strong law for the maximum cumulative sum of independent random variables, Comm. pure appl. math., 18, 109-127, (1965) · Zbl 0135.19205
[12] Hu, Y.; Pierre Loti Viaud, D.; Shi, Z., Laws of the iterated logarithm for iterated Wiener processes, J. theoretic. probab., 8, 303-319, (1995) · Zbl 0816.60027
[13] Hu, Y.; Shi, Z., The Csörgo&#x030B;-Révész modulus of non-differentiability of iterated Brownian motion, Stochastic process. appl., 58, 267-279, (1995) · Zbl 0833.60033
[14] Khoshnevisan, D.; Lewis, T.M., The uniform modulus of iterated Brownian motion, (1994), preprint
[15] Khoshnevisan, D.; Lewis, T.M., Chung’s law of the iterated logarithm for iterated Brownian motion, (1994), preprint · Zbl 0859.60025
[16] Mijnheer, J.L., Sample properties of stable processes, (1975), Mathematisch Centrum Amsterdam · Zbl 0307.60066
[17] Shi, Z., Lower limits of iterated Wiener processes, Statist. probab. lett., 23, 259-270, (1995) · Zbl 0824.60025
[18] Taylor, S.J.; Wendel, J.G., The exact Hausdorff measure of the zero set of stable processes, Z. wahrsch. verw. geb., 6, 37-46, (1967)
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.