Growth and Hölder conditions for the sample paths of Feller processes

Let (A,D(A)) be the infinitesimal generator of a Feller semigroup such that C _c ^∞(ℝ ⁿ )⊂D(A) and A|C _c ^∞(ℝ ⁿ ) is a pseudo-differential operator with symbol −p(x,ξ) satisfying |p(•,ξ)|_∞≤c(1+|ξ|²) and |Imp(x,ξ)|≤c ₀Rep(x,ξ). We show that the associated Feller process {X _t } _{t ≥0} on ℝ ⁿ is a semimartingale, even a homogeneous diffusion with jumps (in the sense of [21]), and characterize the limiting behaviour of its trajectories as t→0 and ∞. To this end, we introduce various indices, e.g., β_∞ ^x :={λ>0:lim _{|ξ|→∞ | x − y |≤2/|ξ|}|p(y,ξ)|/|ξ|^λ=0} or δ_∞ ^x :={λ>0:liminf _{|ξ|→∞ | x − y |≤2/|ξ| |ε|≤1}|p(y,|ξ|ε)|/|ξ|^λ=0}, and obtain a.s. (ℙ ^x ) that lim _{t →0} t ^−1/λ _{s ≤ t} |X _s −x|=0 or ∞ according to λ>β_∞ ^x or λ<δ_∞ ^x . Similar statements hold for the limit inferior and superior, and also for t→∞. Our results extend the constant-coefficient (i.e., Lévy) case considered by W. Pruitt [27]. Received: 21 July 1997 / Revised version: 26 January 1998     

On the Blum-Hanson theorem for quantum quadratic processes

In this paper an analog of the Blum-Hanson theorem for quantum quadratic processes on the von Neumann algebra is proved, i.e., it is established that the following conditions are equivalent:

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:

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:

On the growth of an entire dirichlet series

We establish the relation between the increase of the quantityM(σ,F) = ∣a ₀∣ + ∑ _n=1 ^∞ ∣a _n ∣ exp (σλ _n ) and the behavior of sequences (|a _n |) and (λ _n ), where (λ _n ) is a sequence of nonnegative numbers increasing to + ∞, andF(s) =a ₀ + ∑ _n=1 ^∞ a _n e ^sλn ,s=σ+it, is the Dirichlet entire series. Lviv University, Lviv. Translated from Ukrainskii Matematicheskii Zhurmal, Vol. 51, No. 8, pp. 1149–1153, August, 1999.     

An approximation property of lacunary sequences

Let f ₁ and f ₂ be two positive numbers of the field , and let f _n+2 = f _n+1 + f _n for each n ≥ 1. Let us denote by {x} the fractional part of a real number x. We prove that, for each ξ ∉ K, the inequality {ξf _n } > 2/3 holds for infinitely many positive integers n. On the other hand, we prove a result which implies that there is a transcendental number ξ such that {ξf _n } < 39/40 for each n ≥ 1. Moreover, it is shown that, for every a > 1, there is an interval of positive numbers that contains uncountably many numbers ξ such that {a ⁿ } 6 min 2/(a − 1), (34a ² − 32a + 7)/(9(2a − 1)²) for each n > 1. Here, the minimum is strictly smaller than 1 for each a > 1. In contrast, by an old result of Weyl, for any a > 1, the sequence {ξa ⁿ }, n = 1, 2, ..., is uniformly distributed in [0, 1] (and so everywhere dense in [0, 1]) for almost all real numbers ξ.     

Minimal rates of entropy convergence for completely ergodic systems

If (X,T) is a completely ergodic system, then there exists a positive monotone increasing sequence {a _n } _n ^1/∞ with lim _n →∞a _n =∞ and a positive concave functiong defined on [1, ∞) for whichg(x)/x ² isnot integrable such that for all nontrivial partitions α ofX into two sets.     

Stochastic Equations with Time-Dependent Drift Driven by Levy Processes

The stochastic equation dX _t =dS _t +a(t,X _t )dt, t≥0, is considered where S is a one-dimensional Levy process with the characteristic exponent ψ(ξ),ξ∈ℝ. We prove the existence of (weak) solutions for a bounded, measurable coefficient a and any initial value X ₀=x ₀∈ℝ when (ℛe  ψ(ξ))⁻¹=o(|ξ|⁻¹) as |ξ|→∞. These conditions coincide with those found by Tanaka, Tsuchiya and Watanabe (J. Math. Kyoto Univ. 14(1), 73–92, 1974) in the case of a(t,x)=a(x). Our approach is based on Krylov’s estimates for Levy processes with time-dependent drift. Some variants of those estimates are derived in this note.     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

Atomic Decomposition of Hardy Spaces Associated with Certain Laguerre Expansions

Let L _n ^a (x), n=0,1,…, be the Laguerre polynomials of order a>−1. Denote ℓ _n ^a (x)=(n!/Γ(n+a+1))^1/2 L _n ^a (x)e ^−x/2. Let

Graphs whose every transitive orientation contains almost every relation

Given a graphG onn vertices and a total ordering ≺ ofV(G), the transitive orientation ofG associated with ≺, denotedP(G; ≺), is the partial order onV(G) defined by settingx<y inP(G; ≺) if there is a pathx=x ₁ x ₂…x _r=y inG such thatx ₁ ≺x _j for 1≦i<j≦r. We investigate graphsG such that every transitive orientation ofG contains ₂ ⁿ −o(n ²) relations. We prove that almost everyG _n,p satisfies this requirement if , but almost noG _n,p satisfies the condition if (pn log log logn)/(logn log logn) is bounded. We also show that every graphG withn vertices and at mostcn logn edges has some transitive orientation with fewer than ₂ ⁿ −δ(c)n ² relations. Partially supported by MCS Grant 8104854.     

Good points for diophantine approximation

Given a sequence (x _n ) _n=1^∞ of real numbers in the interval [0, 1) and a sequence (δ _n ) _n=1^∞ of positive numbers tending to zero, we consider the size of the set of numbers in [0, 1] which can be ‘well approximated’ by terms of the first sequence, namely, those y ∈ [0, 1] for which the inequality |y − x _n | < δ _n holds for infinitely many positive integers n. We show that the set of ‘well approximable’ points by a sequence (x _n ) _n=1^∞, which is dense in [0, 1], is ‘quite large’ no matter how fast the sequence (δ _n ) _n=1^∞ converges to zero. On the other hand, for any sequence of positive numbers (δ _n ) _n=1^∞ tending to zero, there is a well distributed sequence (x _n ) _n=1^∞ in the interval [0, 1] such that the set of ‘well approximable’ points y is ‘quite small’.     

On the growth of functions represented by Dirichlet series with complex coefficients on the real axis

We establish conditions under which, for a Dirichlet series F(z) = Σ _{n = 1} ^∞ d _n exp(λ _n z), the inequality ⋎F(x)⋎≤y(x),x≥x _o, implies the relation Σ _{n = 1} ^∞ |d _n exp(λ _n z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞). Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1610–1616. December, 1997     

Asymptotic behaviour of disconnection and non-intersection exponents

Summary. We study the asymptotic behaviour of disconnection and non-intersection exponents for planar Brownian motionwhen the number of considered paths tends to infinity. In particular, if η _n (respectively ξ (n, p)) denotes the disconnection exponent for n paths (respectively the non-intersection exponent for n paths versus p paths), then we show that lim _{n →∞} η _n /n = 1 2 and that for a > 0 and b > 0,lim _{n →∞} ξ ([na],[nb])/n = (√ a + √ b) ² /2. Received: 28 February 1996 / In revised form: 3 September 1996     

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001     

Prime and composite integers close to powers of a number

We prove that, for any real numbers ξ ≠ 0 and ν, the sequence of integer parts [ξ2 ⁿ  + ν], n = 0, 1, 2, . . . , contains infinitely many composite numbers. Moreover, if the number ξ is irrational, then the above sequence contains infinitely many elements divisible by 2 or 3. The same holds for the sequence [ξ( − 2) ⁿ  + ν _n ], n = 0, 1, 2, . . . , where ν ₀, ν ₁, ν ₂, . . . all lie in a half open real interval of length 1/3. For this, we show that if a sequence of integers x ₁, x ₂, x ₃, . . . satisfies the recurrence relation x _n+d  = cx _n  + F(x _n+1, . . . , x _n+d-1) for each n  ≥  1, where c ≠ 0 is an integer, F(z₁,...,z_d-1) ? \mathbb Z[z₁,...,z_d-1],{F(z_1,\dots,z_{d-1}) \in \mathbb {Z}[z_1,\dots,z_{d-1}],} and lim _{n→ ∞}|x _n | = ∞, then the number |x _n | is composite for infinitely many positive integers n. The proofs involve techniques from number theory, linear algebra, combinatorics on words and some kind of symbolic computation modulo 3.     

A general Hobby-Rice Theorem and cake cutting

LetX be a Borel subset of a separable Banach spaceE. Letμ be a non-atomic,σ-finite, Borel measure onX. LetG ⊆L ₁ (X, Σ,μ) bem-dimensional. Theorem:There is an l ∈ E* and real numbers −∞=x ₀<x ₁<x ₂<…<x _n<x _n+1=∞with n≤m, such that for all g ∈ G,      

Possible limit laws for entrance times of an ergodic aperiodic dynamical system

LetG denote the set of decreasingG: ℝ→ℝ withGэ1 on ]−∞,0], and ƒ ₀ ^∞ G(t)dt⩽1. LetX be a compact metric space, andT: X→X a continuous map. Let μ denone aT-invariant ergodic probability measure onX, and assume (X, T, μ) to be aperiodic. LetU⊂X be such that μ(U)>0. Let τ _U (x)=inf{k⩾1:T ^k xεU}, and defineG _U (t)=1/u(U)u({xεU:u(U)τU(x)>t),tεℝ We prove that for μ-a.e.x∈X, there exists a sequence (U _n ) _n≥1 of neighbourhoods ofx such that {x}=∩ _n U _n , and for anyG ∈G, there exists a subsequence (n _k ) _k≥1 withG _{U n k} ↑U weakly. We also construct a uniquely ergodic Toeplitz flowO(x ^∞,S, μ), the orbit closure of a Toeplitz sequencex ^∞, such that the above conclusion still holds, with moreover the requirement that eachU _n be a cylinder set. In memory of Anzelm Iwanik     

Limit theorems for large deviations probabilities of certain quadratic forms

Let (ξ _k ,F _k ) be a martingale difference sequence. The paper concerns the tail behavior of the quadratic formS _n = ∑ _k=1 ⁿ ∑ _j=1 ^k−1 β _n ^k−j χ _k χ _j , where β _n asn→∞. The main conclusions aboutP}n ⁻¹ S _n >x _n }, wherex _n →∞, asn→∞, are obtained using the tail behavior of a martingale with values in a certain Hilbert space. Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 4, pp. 532–549, October–December, 1997.     

Uniform estimates for order statistics and Orlicz functions

We establish uniform estimates for order statistics: Given a sequence of independent identically distributed random variables ξ ₁, … , ξ _n and a vector of scalars x = (x ₁, … , x _n ), and 1 ≤ k ≤ n, we provide estimates for \mathbb E   k-min_{1 ￡ i ￡ n} |x_ix_i|{\mathbb E \, \, k-{\rm min}_{1\leq i\leq n} |x_{i}\xi _{i}|} and \mathbb E k-max_{1 ￡ i ￡ n}|x_ix_i|{\mathbb E\,k-{\rm max}_{1\leq i\leq n}|x_{i}\xi_{i}|} in terms of the values k and the Orlicz norm ||y_x||_M{\|y_x\|_M} of the vector y _x  = (1/x ₁, … , 1/x _n ). Here M(t) is the appropriate Orlicz function associated with the distribution function of the random variable |ξ ₁|, G(t) = \mathbb P ({ |x₁| ￡ t}){G(t) =\mathbb P \left(\left\{ |\xi_1| \leq t\right\}\right)}. For example, if ξ ₁ is the standard N(0, 1) Gaussian random variable, then G(t) = ?{\tfrac2p}ò₀^t e^-\fracs22ds {G(t)= \sqrt{\tfrac{2}{\pi}}\int_{0}^t e^{-\frac{s^{2}}{2}}ds }  and M(s)=?{\tfrac2p}ò₀^se^-\frac12t2dt{M(s)=\sqrt{\tfrac{2}{\pi}}\int_{0}^{s}e^{-\frac{1}{2t^{2}}}dt}. We would like to emphasize that our estimates do not depend on the length n of the sequence.