On Small Deviations of Series of Weighted Random Variables

Let ξ,ξ ₁,ξ ₂,… be positive i.i.d. random variables, S=∑ _j=1^∞ a(j)ξ _j , where the coefficients a(j)≥0 are such that P(S<∞)=1. We obtain an explicit form of the asymptotics of −ln P(S<x) as x→0 for the following three cases:

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

A capacitary method for the asymptotic analysis of dirichlet problems for monotone operators

Given a non-linear elliptic equation of monotone type in a bounded open set Ω ⊂ Rⁿ, we prove that the asymptotic behaviour, asj → ∞, of the solutions of the Dirichlet problems corresponding to a sequence (Ω_j) of open sets contained in Ω is uniquely determined by the asymptotic behaviour, asj → ∞, of suitable non-linear capacities of the setsKΩ _j, whereK runs in the family of all compact subsets of Ω.     

On the bestL 1-approximations of functional classes by splines under restrictions imposed on their derivatives

We find the exact asymptotics (asn→∞) of the bestL ₁-approximations of classesW ₁ ^r of periodic functions by spliness∈S _{2n, r∼-1} (S _{2n, r∼-1} is a set of 2π-periodic polynomial splines of orderr−1, defect one, and with nodes at the pointskπ/n,k∈ℤ) such that V ₀ ^2π s( ^r-1)≤1+ɛ _n , where {ɛ _n } _n=1 ^∞ is a decreasing sequence of positive numbers such that ɛ _n n ²→∞ and ɛ _n →0 asn→∞. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 435–444, April, 1999.     

Norms of powers of absolutely convergent fourier series

Letf(t) = ∑a _k e ^ikt be infinitely differentiable on R, |f(t)|<1. It is known that under these assumptions ‖ⁿ‖ converges to a finite limitl asn → ∞ (l ² = sec(arga),a = (f′(0))² -f″(0)). We obtain here more precise results: (i) an asymptotic series (in powers ofn ^-1/2) for the Fourier coefficientsa _nk off ⁿ , which holds uniformly ink asn → ∞; (ii) an asymptotic series (this time only powers ofn ^-1 are present!) for ‖f ⁿ ‖; (iii) the fact that ifi ^j f ^(j)(0) is real forj = 1,2,..., 2h + 2 then ‖f ⁿ ‖ = l + o(n ^-h ),n → ∞. More generally, we obtain analogous finite asymptotic expansions whenf is assumed to be differentiable only finitely many times.     

Decrease rate of the probabilities ofε-deviations for the means of stationary processes

The asymptotic behavior asn → ∞ of the normed sumsσn =n ⁻¹ Σ _k ⁼⁰ⁿ⁻¹ Xk for a stationary processX = (X _n ,n ∈ ℤ) is studied. For a fixedε > 0, upper estimates for P(sup _k≥n |σ _k | ≥ε) asn → ∞ are obtained. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 366–372, September, 1998.     

Moments of randomly stopped sums-revisited

Conditions are obtained for (*)E|S _T |^γ<∞, γ>2 whereT is a stopping time and {S _n=∑ ₁ ⁿ ,X _j ℱ _n ,n⩾1} is a martingale and these ensure when (**)X _n ,n≥1 are independent, mean zero random variables that (*) holds wheneverET ^γ/2<∞, sup _n≥1 E|X _n |^γ<∞. This, in turn, is applied to obtain conditions for the validity ofE|S _k,T |^γ<∞ and of the second moment equationES _k,T ² =σ ² EΣ _j=k ^T S _k−1,j−1 ² where and {X _n , n≥1} satisfies (**) and ,n≥1. The latter is utilized to elicit information about a moment of a stopping rule. It is also shown for i.i.d. {X _n , n≥1} withEX=0,EX ²=1 that the a.s. limit set of {(n log logn)^−k/2 S _k,n ,n≥k} is [0,2 ^k/2/k!] or [−2 ^k/2/k!] according ask is even or odd and this can readily be reformulated in terms of the corresponding (degenerate kernel)U-statistic .     

L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

A question of Gol’dberg concerning entire functions with prescribed zeros

Let (z_j) be a sequence of complex numbers satisfying |z_j|→ ∞ asj → ∞ and denote by n(r) the number of z_j satisfying |z_j|≤ r. Suppose that lim inf_{r → ⇈} log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫^∞ (ϕ(t)t logt)⁻¹ dt < ∞. It is proved that there exists an entire functionf whose zeros are the z_j such that log log M(r,f) = o((log n(r))²ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.     

Dérivé convexe d’une loi de probabilité et lois stables. Application a un problème d’isomorphisme

The “convex derived set” of a symmetric probability lawF on the real line is defined as the set of limits of laws ∗ _j−1/k ⁿ F(t _j ⁿ η), inf 1≤j≤k _n t _j ⁿ →∞ ifn→∞ and the stable laws it contains are exhibited. A new criterion of stochastic compacity of the set of the powers of a probability law is established. Finally, an isomorphism theorem between somel ^p andL ⁰ spaces is given.

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Laboratoire associé au C.N.R.S. n^o 224 “Processus stochastiques et applications”.     

The average of the values of a function at random points

The behavior of (1/N) asN→∞ is considered, wheref is a bounded measurable function on (−∞, ∞) and (S _n) _n ^=1/∞ are the partial sums of a sequence of independent and identically distributed rondom variables.     

On the series ∑ k=1 ∞ ( k 3k )−1 k −n x k

In this paper we investigate the series ∑ _k=1 ^∞ ( _k ^3k )⁻¹ k ⁻ⁿ x ^k . Obtaining some integral representations of them, we evaluated the sum of them explicitly forn = 0, 1, 2.     

L 1-convergence of double cosine- and Walsh-Fourier series

We prove the convergence inL ¹([−gp, π)²)-norm of the double Fourier series of an integrable functionf(x, y) which is periodic and even with respect tox andy, with coefficientsa _jk satisfying certain conditions of Hardy-Karamata kind, and such thata _jk logj logk→0 asj, k→∞. These sufficient conditions become quite natural in particular cases. Then we extend these results to the convergence of double Walsh-Fourier series inL ¹ (0, 1)²)- norm. As a by-product, we obtain Tauberian conditions ensuring the convergence of a double numerical series provided it is Cesàro summable. This research was partially supported by the Hungarian National Foundation for Scientific Research under Grant # 234.     

On Cramer Approximations under Violation of Cramer's Condition

Let X ₁, X ₂,... be independent identically distributed random variables with distribution function F, S ₀ = 0, S _n = X ₁ + ⋯ + X _n , and Sˉ _n = max_1⩽k⩽n S _k . We obtain large-deviation theorems for S _n and Sˉ _n under the condition 1 − F(x) = P{X ₁ ⩾ x} = e^−l(x), l(x) = x ^α L(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞. __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 447–456, October–December, 2005.     

Transient phenomena for random walks in the absence of the expected value of jumps

Let ξ, ξ₁, ξ₂, ... be independent identically distributed random variables, and S _n :=Σ _j=1 ⁿ ,ξ _j , $ \bar S $ \bar S := sup _n≥0 S _n . If Eξ = −a < 0 then we call transient those phenomena that happen to the distribution $ \bar S $ \bar S as a → 0 and $ \bar S $ \bar S tends to infinity in probability. We consider the case when Eξ fails to exist and study transient phenomena as a → 0 for the following two random walk models:

Embeddings ofl p m intol ∞ n , 1≦p≦2

Isomorphic embeddings ofl _l ^m intol _∞ ⁿ are studied, and ford(n, k)=inf{‖T ‖ ‖T ⁻¹ ‖;T varies over all isomorphic embeddings ofl ₁ ^[klog2n] intol _∞ ⁿ we have that lim _n→∞ d(n, k)=γ(k)⁻¹,k>1, whereγ(k) is the solution of (1+γ)ln(1+γ)+(1 −γ)ln(1 −γ)=k ⁻¹ln4. Here [x] denotes the integer part of the real numberx.     

Commutators in model spaces

Let θ be an inner function, let K _θ = H ² ⊖ θH ², and let S_θ : K_θ → S_θ be defined by the formula S_θf = P_θzf, where f ∈ K_θ is the orthogonal projection of H² onto K_θ. Consider the set A of all trace class operators L : K_θ → K_θ, L = ∑(·,u_n)v_n, ∑∥u_n∥∥v_n∥ < ∞ (u_n, v_n ∈ K_θ), such that ∑ū_n v_n ∈ H ₀ ¹ . It is shown that trace class commutators of the form XS_θ − S_θX (where X is a bounded linear operator on K_θ) are dense in A in the trace class norm. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 333, 2006, pp. 54–61.     

A sieve result for Farey fractions

This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less thanx) lying in an interval (λ₁, λ₂). An asymptotic formula for the sifting function is derived under the assumption that (λ₁, λ₂)x→∞ asx→∞. Two applications of this result are made. In the first one, the value distribution of the vector η(m/n)=(ξ(m), ξ(n)) is considered; here, fork=p ₁ p ₂...p _s ,p ₁≥p ₂>-..., ξk)_is defined by ξ(k)=(logp ₁/logk, logp ₂/logk,..., logp _s /logk, 0, ...); allp _i are prime numbers. It is shown that the limit distribution is π×π, where π is the Poisson-Dirichlet distribution. The asymptotical behavior of finite-dimensional distributions of ξ(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density of sets {m/n: f(m/n)=a}, wheref is a function with the almost squareful kernel. Supported by the Lithuanian State Science and Studies Foundation. Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 108–127, January–March, 1999. Translated by V. Stakénas     

Large deviations probabilities for random walks in the absence of finite expectations of jumps

 Let be independent identically distributed random variables with regularly varying distribution tails: