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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.

60J65 Brownian motion
60J99 Markov processes
60G17 Sample path properties
Full Text: DOI
[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)
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