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.     

Precise Rates in the Law of Iterated Logarithm for the Moment of I.I.D. Random Variables

Let{X,Xn;n≥1} be a sequence of i,i.d, random variables, E X = 0, E X^2 = σ^2 〈 ∞.Set Sn=X1＋X2＋…＋Xn,Mn=max k≤n│Sk│,n≥1.Let an=O（1/loglogn）.In this paper,we prove that,for b〉-1,lim ε→0 →^2（b＋1）∑n=1^∞ （loglogn）^b/nlogn n^1/2 E{Mn-σ（ε＋an）√2nloglogn}＋σ2^-b/（b＋1）（2b＋3）E│N│^2b＋3∑k=0^∞ （-1）k/（2k＋1）^2b＋3 holds if and only if EX=0 and EX^2=σ^2〈∞.     

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,

On extremal properties for the polar derivative of polynomials

If p(z) is a polynomial of degree n having all its zeros on |z| = k, k ≤ 1, then it is proved[5] that max |z|=1 |p′(z)| ≤ kn1n + kn m|z|=ax1 |p(z)|. In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type p(z) = cnzn + ∑n j=μ cn jzn j, 1 ≤μ≤ n. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.     

On the almost sure central limit theorem and domains of attraction

We give necessary and sufficient criteria for a sequence (X _n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,

The area of an exponential random walk and partial sums of uniform order statistics

Let S_i be a random walk with standard exponential increments. The sum ∑ _i=1 ^k S_i is called the k-step area of the walk. The random variable ∑ _i=1 ^k S_i plays an important role in the study of the so-called one-dimensional sticky particles model. We find the distribution of this variable and prove that

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such that an↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞ Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that ∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0 if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.     

Complete moment and integral convergence for sums of negatively associated random variables

For a sequence of identically distributed negatively associated random variables {Xn; n ≥ 1} with partial sums Sn = ∑i=1^n Xi, n ≥ 1, refinements are presented of the classical Baum-Katz and Lai complete convergence theorems. More specifically, necessary and sufficient moment conditions are provided for complete moment convergence of the form ∑n≥n0 n^r-2-1/pq anE（max1≤k≤n｜Sk｜^1/q-∈bn^1/qp）^＋〈∞to hold where r 〉 1, q 〉 0 and either n0 = 1,0 〈 p 〈 2, an = 1,bn = n or n0 = 3,p = 2, an = 1 （log n） ^1/2q, bn=n log n. These results extend results of Chow and of Li and Spataru from the indepen- dent and identically distributed case to the identically distributed negatively associated setting. The complete moment convergence is also shown to be equivalent to a form of complete integral convergence.     

Precise Asymptotics in the Law of the Iterated Logarithm of Moving-Average Processes

In this paper, we discuss the moving-average process Xk = ∑i=-∞ ^∞ ai＋kεi, where {εi;-∞ 〈 i 〈 ∞} is a doubly infinite sequence of identically distributed ψ-mixing or negatively associated random variables with mean zeros and finite variances, {ai;-∞ 〈 i 〈 -∞） is an absolutely solutely summable sequence of real numbers.     

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).     

Almost sure convergence of sample range

Let X ₁, X ₂, ... be i.i.d. random variables. The sample range is R _n = max {X _i , 1 ≤ i ≤ n} − min {X _i , 1 ≤ i ≤ n}. If for a non-degenerate distribution G and some sequences (α _k ), (β _k ) then we have

Asymptotic property of approximation to x αsgn x by Newman Type Operators

Approximation to the function ｜x｜ plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals ｜x｜ if α = 1. We construct a Newman Type Operator rn（x） and prove max ｜x｜≤1｜xαsgn x-rn（x）｜～Cn1/4e-π1/2（1/2）αn.     

Limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival function

LetX ₁,X ₂, ...,X _n be a sequence of nonnegative independent random variables with a common continuous distribution functionF. LetY ₁,Y ₂, ...,Y _n be another sequence of nonnegative independent random variables with a common continuous distribution functionG, also independent of {X _i }. We can only observeZ _i =min(X _i ,Y _i ), and . LetH=1−(1−F)(1−G) be the distribution function ofZ. In this paper, the limit theorems for the ratio of the Kaplan-Meier estimator or the Altshuler estimator to the true survival functionS(t) are given. It is shown that (1)P(δ_(n)=1 i.o.)=0 ifF(τ _H ) < 1 andP(δ _n =0 i.o. )=0 ifG(τH) > 1 where δ_(n) is the corresponding indicator function of and have the same order a.s., where {T _n } is a sequence of constants such that 1−H(T _n )=n ^−α(logn)^β(log logn)^γ.     

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let {Xni} be an array of rowwise negatively associated random variables and Tnk=k∑i=1 i^a Xni for a ≥ -1, Snk =∑｜i｜≤k Ф（i/nη）1/nη Xni for η∈（0,1],where Ф is some function. The author studies necessary and sufficient conditions of ∞∑n=1 AnP（max 1≤k≤n｜Tnk｜〉εBn）〈∞ and ∞∑n=1 CnP（max 0≤k≤mn｜Snk｜〉εDn）〈∞ for all ε 〉 0, where An, Bn, Cn and Dn are some positive constants, mn ∈ N with mn /nη →∞. The results of Lanzinger and Stadtmfiller in 2003 are extended from the i.i.d, case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.     

On the almost sure (a.s.) central limit theorem for random variables with infinite variance

We give criteria for a sequence (X _n ) of i.i.d.r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Asymptotic properties of the norm of the extremum of a sequence of normal random functions

Under additional conditions on a bounded normally distributed random function X = X( t), t ∈ T, we establish a relation of the form

On the Relationship Between the Baum-Katz-Spitzer Complete Convergence Theorem and the Law of the Iterated Logarithm

For a sequence of i.i.d. Banach space-valued random variables {Xn; n ≥ 1} and a sequence of positive constants {an; n ≥ 1}, the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm is investigated. Sets of conditions are provided under which （i） lim sup n→∞ ｜｜Sn｜｜/an〈∞ a.s.and ∞ ∑n=1（1/n）P（｜｜Sn｜｜/an ≥ε〈∞for all ε 〉 λ for some constant λ ∈ [0, ∞） are equivalent; （ii） For all constants λ ∈ [0, ∞）, lim sup ｜｜Sn｜｜/an =λ a.s.and ^∞∑ n=1（1/n） P（｜｜Sn｜｜/an ≥ε）{〈∞, if ε〉λ =∞,if ε〈λare equivalent. In general, no geometric conditions are imposed on the underlying Banach space. Corollaries are presented and new results are obtained even in the case of real-valued random variables.     

Classes of functions subharmonic in ℝm and bounded on certain sets

Let Z_j be the Euclidean space of vectors \((z_{j,1,...,} z_{j_{j \cdot n_j + 1} } ), Z = \mathop \oplus \limits_{j = 1}^P Z_j\) . The function u: Z → ?₊, u ?0, is said to be logarithmically p-subharmonic if log u(z) is upper semicontinuous with respect to the totality of the variables and subharmonic or identically equal to ?∞ with respect to each z_j when the remaining ones are fixed. For such functions, with the growth estimate $$log u(z) \leqslant \delta \mathop \Pi \limits_{j = 1}^P (1 + |z_{j,n_j + 1} |) + N(\mathop {\sum\limits_{\mathop {1 \leqslant j \leqslant p}\limits_{} } {z_{j,k}^2 } }\limits_{1 \leqslant k \leqslant n_j } )^{1/2} + C; \delta ,N \geqslant 0, C \in \mathbb{R}$$ one proves theorems on equivalence of^∞) (L_q)-norms of their restrictions to \(X = \mathop \oplus \limits_{j = 1}^P (Z_{j,1} ,...,z_{j,n_j } )\) and to a relatively dense subset of it, generalizing the known Cartwright and Plancherel-Pólya results.     

Dichotomy laws for the behaviour near the unit element of group representations

A well known “zero-two law" shows that if is a strongly continuous one-parameter group of bounded operators on a Banach space X, and if then Here we discuss analogous problems for general unital representations θ of a topological group G on a unital Banach algebra A. Let 1 be the unit of G, and I the unit element of A. We show that either or if, moreover, θ admits “continuous division by any positive integer”, then, either or Our argument also gives automatic continuity results for representations of abelian Baire groups on a separable Banach algebra and representations of compact non abelian groups on a Banach algebra which are locally bounded and satisfy Received: 8 June 2005; revised: 13 October 2005