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-Strassen approximation scheme , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 ( 1969 ), 321 – 332 . MR 41#1117 [7] K night , F. B. , On the random walk and Brownian

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. [5] C sörgő , M. , Random walking around financial mathematics, R évész , P. (ed.) et al., Random Walks: International workshop, Budapest, Hungary, 1998 . Bolyai Soc. Math. Stud

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Abstract  

Let {X n d }n≥0be a uniform symmetric random walk on Zd, and Π(d) (a,b)={X n d ∈ Zd : a ≤ n ≤ b}. Suppose f(n) is an integer-valued function on n and increases to infinity as n↑∞, and let

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$E_n^{\left( d \right)} = \left\{ {\prod {^{\left( d \right)} } \left( {0,n} \right) \cap \prod {^{\left( d \right)} } \left( {n + f\left( n \right),\infty } \right) \ne \emptyset } \right\}$$ \end{document}
Estimates on the probability of the event
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$E_n^{\left( d \right)}$$ \end{document}
are obtained for
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$d \geqq 3$$ \end{document}
. As an application, a necessary and sufficient condition to ensure
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$P\left( {E_n^{\left( d \right)} ,{\text{i}}{\text{.o}}{\text{.}}} \right) = 0\quad {\text{or}}\quad {\text{1}}$$ \end{document}
is derived for
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$d \geqq 3$$ \end{document}
. These extend some results obtained by Erdős and Taylor about the self-intersections of the simple random walk on Zd.

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504 Miyazaki , H. and Tanaka , H., A theorem of Pitman type for simple random walks on Z d , Tokyo J. Math. , Vol. 12, No. 1. (1989), 235–240. MR 91a :60181

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. , Földes , A. , Révész , P. and Shi , Z. , On the Excursions of twodimensional random walk and Wiener process , Bolyai Society Mathematical Studies , 9 43 – 58 ( 1999 ). [6

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Summary  

In this paper we define Brownian local time as the almost sure limit of the local times of a nested sequence of simple, symmetric random walks. The limit is jointly continuous in \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $(t,x)$ \end{document}. The rate of convergence is \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $n^{\frac14} (\log n)^{\frac34}$ \end{document} that is close to the best possible. The tools we apply are almost exclusively from elementary probability theory.

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Abstract  

The exponential functional of simple, symmetric random walks with negative drift is an infinite polynomial Y = 1 + ξ1 + ξ1ξ2 + ξ1ξ2ξ3 + ⋯ of independent and identically distributed non-negative random variables. It has moments that are rational functions of the variables μk = Ek) < 1 with universal coefficients. It turns out that such a coefficient is equal to the number of permutations with descent set defined by the multiindex of the coefficient. A recursion enumerates all numbers of permutations with given descent sets in the form of a Pascal-type triangle.

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Csáki, E., Földes, A. and Révész, P. , Heavy points of a d-dimensional simple random walk, Statist. Probab. Lett. 76 (2006), 45–57. MR 2006k :60078

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. Bibliografia [1] Csäki , E. , Révész , P. and Shi , Z. , Long excursions of a random walk , J. of Theoretical Probability 14 ( 2001 ), 821 – 844

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Belkin, B. , A limit theorem for conditioned recurrent random walk attracted to a stable law, The Annals of Mathematical Statistics , 41 , No. 1, 146–163, 1970

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