scholarly journals The law of the iterated logarithm on subsequences-characterizations

1990 ◽  
Vol 118 ◽  
pp. 65-97 ◽  
Author(s):  
Michel Weber
Keyword(s):  
Unit Ball ◽  
Covariance Matrix ◽  
Identity Matrix ◽  
Random Vectors ◽  
Unbounded Function ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Euclidian Space ◽  
Zero Vector ◽  
Increasing Sequences

Let be any increasing sequence of integers and M> 1; we connect to them in a very simply way, an increasing unbounded function φ: → R+. Let also X1, X2, · · · be a sequence of i.i.d. random vectors with value in euclidian space Rm. We prove that the cluster set of the sequence almost surely coincides with the unit ball of Rm, if, and only if, the covariance matrix of X1 is the identity matrix of Rm and EX1 is the zero vector of Rm. We define a functional A on the set of increasing sequences of integers as follows:.

scholarly journals Chover-Type Laws of the Iterated Logarithm for Kesten-Spitzer Random Walks in Random Sceneries Belonging to the Domain of Stable Attraction

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Wensheng Wang ◽  
Anwei Zhu
Keyword(s):  
Random Walk ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Law ◽  
Iterated Logarithm ◽  
Cluster Set ◽  
Random Scenery ◽  
Unique Integer

Let X={Xi,i≥1} be a sequence of real valued random variables, S0=0 and Sk=∑i=1kXi  (k≥1). Let σ={σ(x),x∈Z} be a sequence of real valued random variables which are independent of X’s. Denote by Kn=∑k=0nσ(⌊Sk⌋)  (n≥0) Kesten-Spitzer random walk in random scenery, where ⌊a⌋ means the unique integer satisfying ⌊a⌋≤a<⌊a⌋+1. It is assumed that σ’s belong to the domain of attraction of a stable law with index 0<β<2. In this paper, by employing conditional argument, we investigate large deviation inequalities, some sufficient conditions for Chover-type laws of the iterated logarithm and the cluster set for random walk in random scenery Kn. The obtained results supplement to some corresponding results in the literature.

An invariance principle in k-dimensional extended renewal theory

1979 ◽  
Vol 16 (3) ◽  
pp. 567-574 ◽  
Author(s):  
Attila Csenki
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Covariance Matrix ◽  
Invariance Principle ◽  
Exit Time ◽  
Renewal Theory ◽  
Random Vectors ◽  
First Exit Time ◽  
First Passage ◽  
Passage Times

Let ·be a sequence of k -dimensional i.i.d. random vectors and define the first-passage times for where (cvτ)v, τ= 1,· ··,k is the covariance matrix of In this paper the weak convergence of Zn in (D[0, ∞))k is proved under the assumption (0,∞) for all v = 1, ···, k. We deduce the result from the Donsker invariance principle by means of Theorem 5.5 of Billingsley (1968). This method is also used to derive a limit theorem for the first-exit time Mn = min{Nnt for fixed t1,···, tk > 0. The second result is an extension of a theorem of Hunter (1974) whose method of proof applies only if Ρ (ξ1 [0,∞)k) = 1 and μ ν = tv for all v = 1, ···, k.

scholarly journals Second Order Regularity for a Linear Elliptic System Having BMO Coefficients

2021 ◽  
Author(s):  
Gioconda Moscariello ◽  
Giulio Pascale
Keyword(s):  
Dirichlet Problem ◽  
Unit Ball ◽  
Elliptic System ◽  
Positive Constant ◽  
Elliptic Systems ◽  
Second Order ◽  
Identity Matrix ◽  
Bmo Coefficients

AbstractWe consider linear elliptic systems whose prototype is $$\begin{aligned} div \, \Lambda \left[ \,\exp (-|x|) - \log |x|\,\right] I \, Du = div \, F + g \text { in}\, B. \end{aligned}$$ d i v Λ exp ( - | x | ) - log | x | I D u = d i v F + g in B . Here B denotes the unit ball of $$\mathbb {R}^n$$ R n , for $$n > 2$$ n > 2 , centered in the origin, I is the identity matrix, F is a matrix in $$W^{1, 2}(B, \mathbb {R}^{n \times n})$$ W 1 , 2 ( B , R n × n ) , g is a vector in $$L^2(B, \mathbb {R}^n)$$ L 2 ( B , R n ) and $$\Lambda $$ Λ is a positive constant. Our result reads that the gradient of the solution $$u \in W_0^{1, 2}(B, \mathbb {R}^n)$$ u ∈ W 0 1 , 2 ( B , R n ) to Dirichlet problem for system (0.1) is weakly differentiable provided the constant $$\Lambda $$ Λ is not large enough.

scholarly journals Non-Unit Bidiagonal Matrices for Factorization of Vandermonde Matrices

2015 ◽  
Vol 12 ◽  
pp. 1-13
Author(s):  
R. Purushothaman Nair
Keyword(s):  
Vandermonde Matrix ◽  
Inverse Matrix ◽  
Identity Matrix ◽  
Vandermonde Matrices ◽  
The Matrix ◽  
Matrix Transforms ◽  
The Given ◽  
Zero Vector

A non-unit bidiagonal matrix and its inverse with simple structures are introduced. These matrices can be constructed easily using the entries of a given non-zero vector without any computations among the entries. The matrix transforms the given vector to a column of the identity matrix. The given vector can be computed back without any round off error using the inverse matrix. Since a Vandermonde matrix can also be constructed using given n quantities, it is established in this paper that Vandermonde matrices can be factorized in a simple way by applying these bidiagonal matrices. Also it is demonstrated that factors of Vandermonde and inverse Vandermonde matrices can be conveniently presented using the matrices introduced here.

Sign in / Sign up

Export Citation Format

Share Document