Chung’s law of the iterated logarithm for subfractional Brownian motion

Let X~H= {X~H(t), t ∈ R_+} be a subfractional Brownian motion in R~d. We provide a sufficient condition for a self-similar Gaussian process to be strongly locally nondeterministic and show that X~H has the property of strong local nondeterminism. Applying this property and a stochastic integral representation of X~H, we establish Chung's law of the iterated logarithm for X~H.     

Precise asymptotics in Chung’s law of the iterated logarithm

Let X, X1, X2,... be i.i.d, random variables with mean zero and positive, finite variance σ^2, and set Sn = X1 ＋... ＋ Xn, n≥1. The author proves that, if EX^2I{｜X｜≥t} = 0（（log log t）^-1） as t→∞, then for any a〉-1 and b〉 -1,lim ε↑1/√1＋a（1/√1＋a-ε）b＋1 ∑n=1^∞（logn）^a（loglogn）^b/nP{max κ≤n｜Sκ｜≤√σ^2π^2n/8loglogn（ε＋an）}=4/π（1/2（1＋a）^3/2）^b＋1 Г（b＋1）,whenever an = o（1/log log n）. The author obtains the sufficient and necessary conditions for this kind of results to hold.     

Strassen's local law of the iterated logarithm for Lévy's area

We prove a Strassen's law of the iterated logarithm at zero for Lévy's area process. Contrary to the Brownian case, the time inversion argument doesn't seem to work. Here, the main tool in the proof is large deviations estimates for diffusion processes with small diffusion coefficients.     

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati     

On the law of the iterated logarithm for the discrepancy of 〈n k x〉

By a well known result of Philipp (1975), the discrepancy D _N (ω) of the sequence (n _k ω) _k≥1 mod 1 satisfies the law of the iterated logarithm under the Hadamard gap condition n _{k + 1}/n _k ≥ q > 1 (k = 1, 2, …). Recently Berkes, Philipp and Tichy (2006) showed that this result remains valid, under Diophantine conditions on (n _k ), for subexpenentially growing (n _k ), but in general the behavior of (n _k ω) becomes very complicated in the subexponential case. Using a different norming factor depending on the density properties of the sequence (n _k ), in this paper we prove a law of the iterated logarithm for the discrepancy D _N (ω) for subexponentially growing (n _k ) without number theoretic assumptions. C. Aistleitner, Research supported by FWF grant S9603-N13. I. Berkes, Research supported by FWF grant S9603-N13 and OTKA grants K 61052 and K 67961. Authors’ addresses: C. Aistleitner, Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria; I. Berkes, Institute of Statistics, Graz University of Technology, Steyrergasse 17/IV, 8010 Graz, Austria     

The law of the iterated logarithm for local time of a Lévy process

Summary Let {X _t } be a one-dimensional Lévy process with local timeL(t, x) andL ^*(t)=sup{L(t, x): x }. Under an assumption which is more general than being a symmetric stable process with index >1, we obtain a LIL forL*(t). Also with an additional condition of symmetry, a LIL for range is proved.This research is supported by a grant from Korea Science and Engineering Foundations     

Path properties of l_p-valued Gaussian random fields

General limit theorems are established for l~p-valued Gaussian random fields indexed by a multidimensional parameter,which contain both almost sure moduli of continuity and limits of large increments for the l~p-valued Gaussian random fields under(?)explicit conditions.     

An Erdös–Révész type law of the iterated logarithm for reflected fractional Brownian motion

Let \(B_{H}=\{B_{H}(t):t\in \mathbb R\}\) be a fractional Brownian motion with Hurst parameter H ∈ (0,1). For the stationary storage process \(Q_{B_{H}}(t)=\sup _{-\infty <s\le t}(B_{H}(t)-B_{H}(s)-(t-s))\), t ≥ 0, we provide a tractable criterion for assessing whether, for any positive, non-decreasing function f, \( {\mathbb P(Q_{B_{H}}(t) > f(t)\, \text { i.o.})}\) equals 0 or 1. Using this criterion we find that, for a family of functions f _p (t), such that \(z_{p}(t)=\mathbb P(\sup _{s\in [0,f_{p}(t)]}Q_{B_{H}}(s)>f_{p}(t))/f_{p}(t)=\mathcal C(t\log ^{1-p} t)^{-1}\), for some \(\mathcal C>0\), \({\mathbb P(Q_{B_{H}}(t) > f_{p}(t)\, \text { i.o.})= 1_{\{p\ge 0\}}}\). Consequently, with \(\xi _{p} (t) = \sup \{s:0\le s\le t, Q_{B_{H}}(s)\ge f_{p}(s)\}\), for p ≥ 0, \(\lim _{t\to \infty }\xi _{p}(t)=\infty \) and \(\limsup _{t\to \infty }(\xi _{p}(t)-t)=0\) a.s. Complementary, we prove an Erdös–Révész type law of the iterated logarithm lower bound on ξ _p (t), i.e., \(\liminf _{t\to \infty }(\xi _{p}(t)-t)/h_{p}(t) = -1\) a.s., p > 1; \(\liminf _{t\to \infty }\log (\xi _{p}(t)/t)/(h_{p}(t)/t) = -1\) a.s., p ∈ (0,1], where h _p (t) = (1/z _p (t))p loglog t.     

Note on the coefficients of Ramanujan’s expansion for the logarithm of the factorial

Similar to Ramanujan’s expansion for the nth harmonic number, Villarino suggested that there might exist a series expansion for the logarithm of the factorial in terms of the reciprocal of a triangular number. This has been proved in 2010 by Nemes, who gave a complete asymptotic expansion with explicit coefficients and error terms. In this short note, we provide a recursive formula for successively determining the coefficients of the asymptotic expansion by using combinatorial technique.     

Rademacher’s reciprocity law for Dedekind sums in function fields

We consider a Dedekind sum \(s(a,c)\) in function fields, defined via the Carlitz module, similar to the classical Dedekind sum \(D(a,c)\) . In this paper, we prove an analog of Pommersheim’s three-term reciprocity law for \(s(a,c)\) .     

The Hoffmann-Jørgensen inequality of NA random variables and its applications to the logarithm law

In this paper, the Hoffmann-Jørgensen inequality for negatively associated (NA) random variables is derived. As an important tool, it will be applied to the establishing for the logarithm law of NA arrays, and the results of Su, Hu and Liang {xc[15]} are extended.     

Laplace’s method for iterated complex Brownian integrals

A kind of Laplace’s method is developped for iterated stochastic integrals where integrators are complex standard Brownian motions. Then it is used to extend properties of Bougerol and Jeulin’s path transform in the random case when simple representations of complex semisimple Lie algebras are not supposed to be minuscule.     

Precise rate in the law of iterated logarithm for <Emphasis Type="Italic">ρ</Emphasis>-mixing sequence

Let {X, X _n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim _n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,

A proof of Price’s law for the collapse of a self-gravitating scalar field

A well-known open problem in general relativity, dating back to 1972, has been to prove Price’s law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price’s law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.’s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author concerning the internal structure of charged black holes, which had assumed the validity of Price’s law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou’s C⁰-formulation, the conjecture is proven to be false.     

Sufficient conditions for Benford’s law

We present two sufficient conditions for an absolutely continuous random variable to obey Benford’s law for the distribution of the first significant digit. These two sufficient conditions suggest that Benford’s law will not often be observed in everyday sets of numerical data. On the other hand, we recall that there are two processes by way of which a random variable can come close to following Benford’s law. The first of these is the multiplication of independent random variables and the second is the exponentiation of a random variable to a large power. Our working tool is the Poisson sum formula of Fourier analysis. Like the central limit theorem, Benford’s law has an asymptotic nature.     

The geometry of the Wilks’s Λ random field

The statistical problem addressed in this paper is to approximate the P value of the maximum of a smooth random field of Wilks’s Λ statistics. So far results are only available for the usual univariate statistics (Z, t, χ², F) and a few multivariate statistics (Hotelling’s T ², maximum canonical correlation, Roy’s maximum root). We derive results for any differentiable scalar function of two independent Wishart random fields, such as Wilks’s Λ random field. We apply our results to a problem in brain shape analysis.     

Burnside’s theorem in the setting of general fields

Archiv der Mathematik - We study the mean of the values of the zeta-function on a generalized arithmetic progression on the critical line.