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Joint asymptotic behavior of local and occupation times of random walk in higher dimension

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 44: Issue 4
DOI:
https://doi.org/10.1556/sscmath.2007.1035
Authors: Endre Csáki, Antónia Földes, and Pál Révész

Considering a simple symmetric random walk in dimension d ≧ 3, we study the almost sure joint asymptotic behavior of two objects: first the local times of a pair of neighboring points, then the local time of a point and the occupation time of the surface of the unit ball around it.

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A functional CLT for the L 2 modulus of continuity of local time

Periodica Mathematica Hungarica
Volume/Issue:
Volume 61: Issue 1-2
DOI:
https://doi.org/10.1007/s10998-010-3231-8
Author: Jay Rosen

Abstract  

We show that as processes in (c, d, t) ∈ C(R 2 × R + 1)

\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} $$\frac{{\int_c^d {(L_t^{x + h} - L_t^x )^2 dx - 4h} \int_c^d {L_t^x dx} }} {{h^{3/2} }}\mathop \Rightarrow \limits^\mathcal{L} \left( {\frac{{64}} {3}} \right)^{1/2} \int_c^d {L_t^x d\eta (x)}$$ \end{document}
as h → 0 for Brownian local time L t x. Here η(x) is an independent two-sided Brownian motion.

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Some extensions of Pitman and Ray-Knight theorems for penalized Brownian motions and their local times, IV

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 44: Issue 4
DOI:
https://doi.org/10.1556/sscmath.2007.1032
Authors: Bernard Roynette, Pierre Vallois, and Marc Yor

We show that Pitman’s theorem relating Brownian motion and the BES (3) process, as well as the Ray-Knight theorems for Brownian local times remain valid, mutatis mutandis, under the limiting laws of Brownian motion penalized by a function of its one-sided maximum.

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Penalizing a BES ( d ) process (0 < d < 2) with a function of its local time, V

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 45: Issue 1
DOI:
https://doi.org/10.1556/sscmath.2007.1042
Authors: Bernard Roynette, Pierre Vallois, and Marc Yor

We describe the limit laws, as t → ∞, of a Bessel process ( R s , s ≦ t ) of dimension d ∈ (0, 2) penalized by an integrable function of its local time L t at 0, thus extending our previous work of this kind, relative to Brownian motion.

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Random walk and Brownian local times in Wiener sheets: A tribute to my almost surely most visited 75 years young best friends, Endre Csáki and Pál Révész

Periodica Mathematica Hungarica
Volume/Issue:
Volume 61: Issue 1-2
DOI:
https://doi.org/10.1007/s10998-010-3001-7
Author: Miklós Csörgő
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