Upper and lower class separating sequences for brownian motion with random argument

• Autorzy: Aistleitner C. , Hörmann S.
• Języki publikacji: EN

Abstrakty

EN

Let X = X₁,X₂, . . . be a sequence of random variables, let W be a Brownian motion independent of X and let Z_k = W(X_k). A numerical sequence (tk) will be called an upper (lower) class sequence for {Z_k} if P(Z_k > t_k for infinitely many k) = 0 (or 1, respectively). At a first look one might be tempted to believe that a “separating line” (t⁰_k), say, between the upper and lower class sequences for {Z_k} is directly related to the corresponding counterpart (s^{0k) for the process {Xk}. For example, by using the law of the iterated logarithm for the Wiener process a functional relationship t0}_k = √2s⁰_k log log s⁰_k seems to be natural. If X_k = |W₂(k)| for a second Brownian motion W₂ then we are dealing with an iterated Brownian motion, and it is known that the multiplicative constant √2 in (0.1) needs to be replaced by 2 · 3^−3/4, contradicting this simple argument. We will study this phenomenon from a different angle by letting {X_k} be an i.i.d. sequence. It turns out that the relationship between the separating sequences (s⁰_k) and (t⁰_k) in the above sense depends in an interesting way on the extreme value behavior of {X_k}.

Słowa kluczowe

EN

Brownian motion; extreme values; iterated Brownian motion; upper-lower class test

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2011
• Tom: Vol. 31, Fasc. 2
• Strony: 183--202
• Opis fizyczny: Bibliogr. 11 poz.

Twórcy

autor

Aistleitner C.

• Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

autor

Hörmann S.

• Département de Mathématique, Université Libre de Bruxelles, CP 210, Bd du Triomphe, 1050 Brussels, Belgium

Bibliografia

• [1] K. Burdzy, Some path properties of iterated Brownian motion, in: Seminar on Stochastic Processes, 1992 (Seattle, WA, 1992), Progr. Probab. 33 (1993), pp. 67-87.
• [2] E. Csáki, M. Csörgő, A. Földes and P. Révész, Brownian local time approximated by a Wiener sheet, Ann. Probab. 17 (1989), pp. 516-537.
• [3] E. Csáki, M. Csörgő, A. Földes and P. Révész, Global Strassen-type theorems for iterated Brownian motions, Stochastic Process. Appl. 59 (1995), pp. 321-341.
• [4] W. Feller, The general form of the so-called law of the iterated logarithm, Trans. Amer. Math. Soc. 54 (1943), pp. 373-402.
• [5] W. Feller, The law of the iterated logarithm for identically distributed random variables, Ann. of Math. 47 (1946), pp. 631-638.
• [6] Y. Hu, D. Pierre-Loti-Viaud and Z. Shi, Laws of the iterated logarithm for iterated Wiener processes, J. Theoret. Probab. 8 (1995), pp. 303-319.
• [7] D. Khoshnevisan and T. M. Lewis, Chung’s law of the iterated logarithm for iterated Brownian motion, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), pp. 349-359.
• [8] S. Kochen and C. Stone, A note on the Borel-Cantelli lemma, Illinois J. Math. 8 (1964), pp. 248-251.
• [9] M.R. Leadbetter, G. Lindgren and H. Rootzén, Extremes and Related Properties of Random Sequences and Processes, Springer Ser. in Statist., New York 1983.
• [10] V.V. Petrov, A generalization of the Borel-Cantelli lemma, Statist. Probab. Lett. 67 (2004), pp. 233-239.
• [11] F. Spitzer, Principles of Random Walk, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London 1964.