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Moderate deviations and law of the iterated logarithm for intersections of the ranges of random walks  [PDF]
Xia Chen
Mathematics , 2005, DOI: 10.1214/009117905000000035
Abstract: Let S_1(n),...,S_p(n) be independent symmetric random walks in Z^d. We establish moderate deviations and law of the iterated logarithm for the intersection of the ranges #{S_1[0,n]\cap... \cap S_p[0,n]} in the case d=2, p\ge 2 and the case d=3, p=2.
Moderate deviations and laws of the iterated logarithm for the local times of additive Lévy processes and additive random walks  [PDF]
Xia Chen
Mathematics , 2007, DOI: 10.1214/009117906000000520
Abstract: We study the upper tail behaviors of the local times of the additive L\'{e}vy processes and additive random walks. The limit forms we establish are the moderate deviations and the laws of the iterated logarithm for the L_2-norms of the local times and for the local times at a fixed site.
Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm  [PDF]
Richard F. Bass,Xia Chen
Mathematics , 2005, DOI: 10.1214/009117904000000504
Abstract: If \beta_t is renormalized self-intersection local time for planar Brownian motion, we characterize when Ee^{\gamma\beta_1} is finite or infinite in terms of the best constant of a Gagliardo-Nirenberg inequality. We prove large deviation estimates for \beta_1 and -\beta_1. We establish lim sup and lim inf laws of the iterated logarithm for \beta_t as t\to\infty.
Large deviations and laws of the iterated logarithm for the local times of additive stable processes  [PDF]
Xia Chen
Mathematics , 2006, DOI: 10.1214/009117906000000601
Abstract: We study the upper tail behaviors of the local times of the additive stable processes. Let $X_1(t),...,X_p(t)$ be independent, d-dimensional symmetric stable processes with stable index $0<\alpha\le 2$ and consider the additive stable process $\bar{X}(t_1,...,t_p)=X_1(t_1)+... +X_p(t_p)$. Under the condition $d<\alpha p$, we obtain a precise form of the large deviation principle for the local time \[\eta^x([0,t]^p)=\int_0^t...\int_0^t\delta_x\bigl(X_1(s_1)+... +X_p(s_p)\bigr) ds_1... ds_p\] of the multiparameter process $\bar{X}(t_1,...,t_p)$, and for its supremum norm $\sup_{x\in\mathbb{R}^d}\eta^x([0,t]^p)$. Our results apply to the law of the iterated logarithm and our approach is based on Fourier analysis, moment computation and time exponentiation.
General Large Deviations and Functional Iterated Logarithm Law for Multivalued Stochastic Differential Equations  [PDF]
Jiagang Ren,Jing Wu,Hua Zhang
Mathematics , 2015,
Abstract: In this paper, we prove a large deviation principle of Freidlin-Wentzell's type for the multivalued stochastic differential equations. As an application, we derive a functional iterated logarithm law for the solutions of multivalued stochastic differential equations.
Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation  [PDF]
Li-Xin Zhang
Mathematics , 2015,
Abstract: The sub-linear expectation or called G-expectation is a nonlinear expectation having advantage of modeling non-additive probability problems and the volatility uncertainty in finance. Let $\{X_n;n\ge 1\}$ be a sequence of independent random variables in a sub-linear expectation space $(\Omega, \mathscr{H}, \widehat{\mathbb E})$. Denote $S_n=\sum_{k=1}^n X_k$ and $V_n^2=\sum_{k=1}^n X_k^2$. In this paper, a moderate deviation for self-normalized sums, that is, the asymptotic capacity of the event $\{S_n/V_n \ge x_n \}$ for $x_n=o(\sqrt{n})$, is found both for identically distributed random variables and independent but not necessarily identically distributed random variables. As an applications, the self-normalized laws of the iterated logarithm are obtained.
Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks  [PDF]
Xia Chen
Mathematics , 2005, DOI: 10.1214/009117904000000513
Abstract: Let \alpha ([0,1]^p) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d-2)
Chover-Type Laws of the Iterated Logarithm for Continuous Time Random Walks
Kyo-Shin Hwang,Wensheng Wang
Journal of Applied Mathematics , 2012, DOI: 10.1155/2012/906373
Abstract: A continuous time random walk is a random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper, we establish Chover-type laws of the iterated logarithm for continuous time random walks with jumps and waiting times in the domains of attraction of stable laws.
Law of the Iterated Logarithm for the random walk on the infinite percolation cluster  [PDF]
H. Duminil-Copin
Mathematics , 2008,
Abstract: We show that random walks on the infinite supercritical percolation clusters in Z^d satisfy the usual Law of the Iterated Logarithm. The proof combines Barlow's Gaussian heat kernel estimates and the ergodicity of the random walk on the environment viewed from the random walker as derived by Berger and Biskup.
The Law of the Iterated Logarithm in Analaysis  [PDF]
Santosh Ghimire
Journal of the Institute of Engineering , 2013, DOI: 10.3126/jie.v9i1.10674
Abstract: ?In this paper, we first discuss the history of the law of the iterated logarithm. We then focus our discussion on how it was introduced in analysis. Finally we mention different types of law of the iterated logarithm and state some of the recent developments. DOI: http://dx.doi.org/10.3126/jie.v9i1.10674 Journal of the Institute of Engineering , Vol. 9, No. 1, pp. 89–94
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