ON THE GROWTH OF SOME RANDOM HYPERDIRICHLET SERIES

The paper considers the random L-Dirichlet seriesf(s,ω)=sum from n=1 to ∞ P_n(s,ω)exp(-λ_ns)and the random B-Dirichlet seriesψτ_0(s,ω)=sum from n=1 to ∞ P_n(σ iτ_0,ω)exp(-λ_ns),where {λ_n} is a sequence of positive numbers tending strictly monotonically to infinity, τ_0∈R is a fixed real number, andP_n(s,ω)=sum from j=1 to m_n ε_(nj)a_(nj)s~ja random complex polynomial of order m_n, with {ε_(nj)} denoting a Rademacher sequence and {a_(nj)} a sequence of complex constants. It is shown here that under certain very general conditions, almost all the random entire functions f(s,ω) and ψ_(τ_0)(s,ω) have, in every horizontal strip, the same order, given byρ=lim sup((λ_nlogλ_n)/(log A_n~(-1)))whereA_n=max |a_(nj)|.Similar results are given if the Rademacher sequence {ε_(nj)} is replaced by a steinhaus seqence or a complex normal sequence.     

ON POINTS CONTAIN ARITHMETIC PROGRESSIONS IN THEIR LROTH EXPANSION

For any x ∈(0, 1](except at most countably many points), there exists a unique sequence {dn(x)}n≥1of integers, called the digit sequence of x, such that ■.The dexter infinite series expansion is called the L¨uroth expansion of x. This paper is concerned with the size of the set of points x whose digit sequence in its L¨uroth expansion is strictly increasing and contains arbitrarily long arithmetic progressions with arbitrary common difference. More precisely, we determine the Hausdorff dimension of the above set.     

ON CHOVER''''S LIL FOR THE WEIGHTED SUMS OF STABLE RANDOM VARIABLES

Let {Xn,n ≥ 1} be a sequence of α-stable random variables(0 < α < 2), {ani,1 ≤ i≤ n, n≥1} be an array of constant real numbers. Under some restriction of {ani,1 ≤ i ≤ n,n≥1}, the authors discuss the integral test for the weighted partial sums {Σi=1naniXi,n ≥ 1}, and obtain the Chover's laws of iterated logarithm(LIL) as corollaries.     

THE NATURAL BOUNDARY OF SOME RANDOM POWER SERIES

Suppose that {X_n(ω)} are independent random complex variable sequence, E(X_n)-0 and Then the circle {|Z|=ρ} is almost surely a natural boundary of the random series sum from n=1 to ∞ X_n(ω)Z~(n-1)     

一类变时滞差分方程的振动性

Consider the nonautonomous delay logistic equation △yn=pnyn(1-n≥0 (1)where {pn}n≥0 is a sequence of nonnegative real numbers, {ln}n≥0 is a sequence of positive integers satisfying lim (n-ln)=∞, and k is a positive constant. Only solutions which are positive for n ≥ 0 are considered. We obtain a new sufficient for all positive solutions of (1) to oscillate about k which contains the corresponding result in [2] when I=1.     

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.     

COMPARISON THEOREMS AND OSCILLATION CRITERIA FOR DIFFERENCE EQUATIONS WITH SEVERAL DELAYS

In this paper, we first establish the equivalence of the oscillation of thedifference equations with several delays of the form: Δx_n+sum from i=1 to m(pi(n)x_(n-k_i))=0 for n≥0and the second-order difference equations without delay of the form:where{pj(n)}is a sequence of nonnegative real numbers and{k_i}_(i=1)~m is a setof positive integers. Then we get some "sharp" conditions for oscillation andnon-oscillation of the first equation.     

Proofs of some conjectures on monotonicity of number-theoretic and combinatorial sequences

In 2012,Zhi-Wei Sun posed many conjectures about the monotonicity of sequences of form {n√zn},where {zn} is a familiar number-theoretic or combinatorial sequence. We show that if the sequence {zn+1/zn}is increasing(resp.,decreasing),then the sequence {n√zn} is strictly increasing(resp.,decreasing) subject to a certain initial condition. We also give some sufficient conditions when {zn+1/zn} is increasing,which is equivalent to the log-convexity of {zn}. As consequences,a series of conjectures of Zhi-Wei Sun are verified in a unified approach.     

ON THE PROBLEM OF BEST CONVERGENCE RATES OF DENSITY ESTIMATES

Let X_1,…,X_n be iid samples drawn from an m-dimensional population with a probabilitydensity f,belonging to the family C_(ka),i.e.the family of all densities whose partialderivatives of order k are bounded by a.It is desired to estimate the value of f at somepredetermined point a,for example a=0.Farrell obtained some results concerning the bestpossible convergence rates for all estimator sequence,from which it follows,for example,thatthere exists no estimator sequence{γ_n(0)=γ_n(X_1,…,X_n,0)}such that(?)E_f[γ_n(0)-f(0)]~2=o(n~(-2k/(2k m))).This article pursues this problem further and proves that there existsno estimator sequence{γ_n(0)}such thatn~(-k/(2k m))(γ_n(0)-f(0))(?)0,for each f∈C_(ka),where(?)denotes convergence in probability.     

A nonclassical LIL for geometrically weighted series in Banach space

Let {X, Xn ; n ≥ 0} be a sequence of independent and identically distributed random variables, taking values in a separable Banach space (B,||·||) with topological dual B* . Considering the geometrically weighted series ξ(β) =∑∞n=0βnXn for 0 β 1, and a sequence of positive constants {h(n), n ≥ 1}, which is monotonically approaching infinity and not asymptotically equivalent to log log n, a limit result for(1-β2)1/2||ξ(β)||/(2h(1/(1-β2)))1/2 is achieved.     

MEAN APPROXIMATION OF A PERIODIC FUNCTION BY THE PARTIAL SUMS OF ITS FOURIER SERIES

Given a sequence of positive real numbers \[{\varepsilon _0},{\varepsilon _1},...,{\varepsilon _n},...\] which satisfy the conditions \[{\varepsilon _v} \to 0,{\varepsilon _v} - {\varepsilon _{v + 1}} \ge 0,{\varepsilon _v} - 2{\varepsilon _{v + 1}} + {\varepsilon _{v + 2}} \ge 0\] for v =0, 1, 2, ..., and a class L(s) of 2pi-periodic, L-integrable functions f(x) such that \[{E_n}{(f)_L} \le {\varepsilon _n}(n = 0,1,2,...)\], where \[{E_n}{(f)_L}\] is the best mean approximation of f(x) by trigonometrical polynomials of degree ≤n Let \[{S_n}(f)\] be the n-th partial sum of the Fourier series of f(x). It’s known that Oskolkov has proved \[\mathop {\sup }\limits_{f \in L(\varepsilon )} ||f - {S_n}{(f)_L}|| = \sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} \] where \[||f|{|_L} = \int_0^{2\pi } {|f(x)|} dx\] Oskolkov asked whether there is a single function \[{f_0}(x) \in L(s)\] for which the above relation is satisfied for all n, In this paper the following result is obtained. Theorem Let \[L(\varepsilon )\] be a class of 2pi-periodic, L-integrable functions as giyen above, then there exists a funotion \[{f_0}(x) \in L(\varepsilon )\] such that \[{{\tilde f}_0}(x) \in L(\varepsilon )\] and \[\begin{array}{l} \overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{{{\left\| {{f_0} - {S_n}({f_0})} \right\|}_L}}}{{\sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} }} \ge C > 0\\overline {\mathop {\lim }\limits_{n \to \infty } } \frac{{{{\left\| {{{\tilde f}_0} - {S_n}({{\tilde f}_0})} \right\|}_L}}}{{\sum\limits_{v = n}^{2n} {\frac{{{\varepsilon _n}}}{{v - n + 1}}} }} \ge C > 0 \end{array}\] where C is an absolute constant. Some generalizations of the theorem are given.     

On the convergence for PNQD sequences with general moment conditions

Let {X,X_n,n≥ 1} be a sequence of identically distributed pairwise negative quadrant dependent(PNQD) random variables and {a_n,n≥ 1} be a sequence of positive constants with a_n=f(n) and f(θ~k)/f(θ~(k-1)≥β for all large positive integers k, where 1 θ≤β and f(x) 0(x≥1) is a non-decreasing function on [b,+∞) for some b≥1.In this paper,we obtain the strong law of large numbers and complete convergence for the sequence {X,X_n, n≥ 1},which are equivalent to the general moment condition∑_(n=1)~∞ P(|X| a_n) ∞.Our results extend and improve the related known works in Baum and Katz [1],Chen at al.[3],and Sung[14].     

Precise Rates in the Generalized Law of the Iterated Logarithm in R~m

Let {X, X_n, n ≥ 1} be a sequence of i.i.d. random vectors with EX =(0,..., 0)_(m×1) and Cov(X, X) = σ~2 Ⅰ_m, and set S_n =∑_(i=1)~n X_i, n ≥ 1. For every d 0 and a_n =o((log log n)~(-d)), the article deals with the precise rates in the genenralized law of the iterated logarithm for a kind of weighted infinite series of P(|S_n| ≥(ε + a_n)σn~(1/2)(log log n)~d).     

The Unitary Equivalence Problem of Unilateral Weighted Shifts

In this paper, we study the unilateral weighted shift which is unitarily equivalent to a Toeplitz operator and prove a similar result to that in [1] without the hypothesis that the shift must be hyponormal. As a corollary, we show that if the weight sequence {α_n}_(n=0)~∞ of the shift is convergent, then 1-a_n~2=(1-a_0~2)~(n 1) n≥0     

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.     

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.     

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.     

The precise asymptotics of the complete convergence for moving average processes of <Emphasis Type="Italic">m</Emphasis>-dependent <Emphasis Type="Italic">B</Emphasis>-valued elements

Let {εt;t ∈ Z} be a sequence of m-dependent B-valued random elements with mean zeros and finite second moment. {a3;j ∈ Z} is a sequence of real numbers satisfying ∑j=-∞^∞｜aj｜ 〈 ∞. Define a moving average process Xt = ∑j=-∞^∞aj＋tEj,t ≥ 1, and Sn = ∑t=1^n Xt,n ≥ 1. In this article, by using the weak convergence theorem of { Sn/√ n _〉 1}, we study the precise asymptotics of the complete convergence for the sequence {Xt; t ∈ N}.     

ON ALMOST SURE CONVERGENCE OF WEIGHTED SUMS OF RANDOM ELEMENT SEQUENCES

We mainly study the almost sure limiting behavior of weighted sums of the form ∑ni=aiXi/bn , where {Xn, n ≥ 1} is an arbitrary Banach space valued random element sequence or Banach space valued martingale difference sequence and {an, n ≥ 1} and {bn,n ≥ 1} are two sequences of positive constants. Some new strong laws of large numbers for such weighted sums are proved under mild conditions.     

STRONG APPROXIMATION FOR MOVING AVERAGE PROCESSES UNDER DEPENDENCE ASSUMPTIONS

Let {Xt,t ≥ 1} be a moving average process defined by Xt = ∑^∞ k=0 αkξt-k, where {αk,k ≥ 0} is a sequence of real numbers and {ξt,-∞ 〈 t 〈 ∞} is a doubly infinite sequence of strictly stationary dependent random variables. Under the conditions of {αk, k ≥ 0} which entail that {Xt, t ≥ 1} is either a long memory process or a linear process, the strong approximation of {Xt, t ≥ 1} to a Gaussian process is studied. Finally, the results are applied to obtain the strong approximation of a long memory process to a fractional Brownian motion and the laws of the iterated logarithm for moving average processes.