Precise asymptotics in the self-normalized law of the iterated logarithm

Let X,X₁,X₂,… be i.i.d. nondegenerate random variables with zero means, and . We investigate the precise asymptotics in the law of the iterated logarithm for self-normalized sums, S_n/V_n, also for the maximum of self-normalized sums, max_1kn|S_k|/V_n, when X belongs to the domain of attraction of the normal law.     

A supplement to the Davis–Gut law

Let {X,X_n;n1} be a sequence of i.i.d. real-valued random variables and set , n1. Let h() be a positive nondecreasing function such that . Define Lt=log_emax{e,t} for t0. In this note we prove that

if and only if E(X)=0 and E(X²)=1, where , t1. When h(t)≡1, this result yields what is called the Davis–Gut law. Specializing our result to h(t)=(Lt)^r, 0<r1, we obtain an analog of the Davis–Gut law.     

Precise asymptotics in the law of the iterated logarithm*

Let X₁, X₂, ... be i.i.d. random variables with EX₁ = 0 and positive, finite variance σ², and set S_n = X₁ + ... + X_n. For any α > −1, β > −1/2 and for κ_n(ε) a function of ε and n such that κ_n(ε) log log n → λ as n ↑ ∞ and , we prove that

Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let be a sequence of i.i.d. random variables with EX=0 and EX²=σ²<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain

     

Laws of the single logarithm for delayed sums of random fields II

In an earlier paper we extended Lai's (1974) law of the single logarithm for delayed sums to a class of delayed sums of random fields. In this paper we allow for more general windows.     

Some asymptotic results related to the law of iterated logarithm for Brownian motion

For 0<<1, let . The questions addressed in this paper are motivated by a result due to Strassen: almost surely, lim sup _t U ((t))=1–exp{–4(^–1)–1}. We show that Strassen's result is closely related to a large deviations principle for the family of random variablesU (t), t>0. Also, when =1,U (t)0 almost surely and we obtain some bounds on the rate of convergence. Finally, we prove an analogous limit theorem for discounted averages of the form as 0, whereD is a suitable discount function. These results also hold for symmetric random walks.     

On the rates of the other law of the logarithm

Let X, X1 , X2 , . . . be i.i.d. random variables, and set Sn = X1 +···+Xn , Mn = maxk≤n |Sk|, n ≥1. Let an = o( (n)(1/2)/logn). By using the strong approximation, we prove that, if EX = 0, VarX = σ2 0 and E|X| 2+ε ∞ for some ε 0, then for any r 1, lim ε1/(r-1)(1/2) [ε-2-(r-1)]∞∑n=1 nr-2 P{Mn ≤εσ (π2n/(8log n))(1/2) + an } = 4/π . We also show that the widest a n is o( n(1/2)/logn).     

Precise rates in the law of the logarithm in the Hilbert space

Let be a sequence of i.i.d. random variables taking values in a real separable Hilbert space (H,‖⋅‖) with covariance operator Σ, and set Sn=X₁+?+Xn, n?1. Let . We prove that, for any 1<r<3/2 and a>−d/2,

     

Precise asymptotics for a series of T. L. Lai

Let be i.i.d. random variables with , and set . We prove that, for

under the assumption that and Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974).

     

A functional law of the iterated logarithm for associated sequences

Recently, a functional central limit theorem and a Berry-Essen Theorem have been demonstrated for classes or associated random variables. Using these results, and similar results for multiplicative sequences, we show a functional law of the iterated logarithm for associated sequences satisfying a rate requirement.     

A law of the iterated logarithm for arithmetic functions

Let be a sequence of centered iid random variables. Let be a strongly additive arithmetic function such that and put . If and satisfies a Lindeberg-type condition, we prove the following law of the iterated logarithm:

We also prove the validity of the corresponding weighted strong law of large numbers in .

     

A note on some laws of the iterated logarithm

Some function space laws of the iterated logarithm for Brownian motion with values in finite and infinite dimensional vector spaces are shown to follow from Hincin's classical law of the iterated logarithm and some martingale techniques. A law of the iterated logarithm for Brownian motion in a differentible manifold is also stated.     

On the Makarov law of the iterated logarithm

We obtain considerable improvement of Makarov's estimate of the boundary behavior of a general conformal mapping from the unit disk to a simply connected domain in the complex plane. We apply the result to improve Makarov's comparison of harmonic measure with Hausdorff measure on simply connected domains.

     

负相伴随机变量的非经典重对数律

§ 1　 IntroductionA finite family of random variables { Xi,1≤ i≤ n} is said to be negatively associated(NA) is for every pair of disjointsubsets A1 and A2 of{ 1 ,2 ,...,n} ,Cov{ f1 (Xi,i∈ A1 ) ,f2 (Xj,j∈ A2 ) }≤ 0 ,(1 .1 )whenever f1 and f2 are coordinatewise increasing and the covariance exists.An infinitefamily is negatively associated ifevery finite subfamily is negatively associated.This defini-tion was introduced by Alam and Saxena[1 ] and Joag-Dev and Proschan[2 ] .As pointed…     

Precise asymptotics in some strong limit theorems for multidimensionally indexed random variables

Consider Z₊^d (d2)—the positive d-dimensional lattice points with partial ordering , let {X_k,kZ₊^d} be i.i.d. random variables with mean 0, and set S_n=∑_knX_k, nZ₊^d. We establish precise asymptotics for ∑_n|n|^r/p−2P(|S_n||n|^1/p), and for

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, (0δ1) as 0, and for

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as .     

Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

A law of the iterated logarithm for the product limit estimator with doubly censored data

We establish the law of the iterated logarithm for the product limit estimator, when the data are subject to double censoring. This investigation extends the results available for the model for singly censored data.     

A law of the iterated logarithm for Markov chains on ℕ0 associated with orthogonal polynomials

We derive laws of the iterated logarithm for Markov chains on the nonnegative integers whose transition probabilities are associated with a sequence of orthogonal polynomials. These laws can be applied to a large class of birth and death random walks and random walks on polynomial hypergroups. In particular, the results of our paper lead immediately to a law of the iterated logarithm for the growth of the distance of isotropic random walks on infinite distance-transitive graphs as well as on certain finitely generated semigroups from their starting points.     

A law of the iterated logarithm for processes with independent increments

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On the law of the iterated logarithm in open queueing networks

An open queueing network model in heavy traffic is developed. Such models are mathematical models of computer networks in heavy traffic. Laws of the iterated logarithm for the virtual waiting time of the customer in open queueing networks and homogeneous computer networks are proved.