Stochastic approximation with dependent noise

In this work we derive the usual limit laws (weak and strong convergence, central limit theorem, invariance principle) for stochastic approximation with stationary noise. The idea is to introduce an artificial sequence, related to the SA scheme, but which clearly obeys the desired limit law. This sequence is subtracted from the SA scheme and the remainder, which behaves more or less deterministically, is shown to vanish using simple limit arguments.     

Some limit theorems for negatively associated random variables

Let {X _n , n ≥ 1} be a sequence of negatively associated random variables. The aim of this paper is to establish some limit theorems of negatively associated sequence, which include the L ^p -convergence theorem and Marcinkiewicz–Zygmund strong law of large numbers. Furthermore, we consider the strong law of sums of order statistics, which are sampled from negatively associated random variables.     

Benford’s law and distribution functions of sequences in (0, 1)

Applying the theory of distribution functions of sequences x _n ∈ [0, 1], n = 1, 2, ..., we find a functional equation for distribution functions of a sequence x _n and show that the satisfaction of this functional equation for a sequence x _n is equivalent to the fact that the sequence x _n to satisfies the strong Benford law. Examples of distribution functions of sequences satisfying the functional equation are given with an application to the strong Benford law in different bases. Several direct consequences from uniform distribution theory are shown for the strong Benford law.     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

Devaney’s chaos on uniform limit maps

Let (X, d) be a compact metric space and f_n : X → X a sequence of continuous maps such that (f_n) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all f_n mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1]. On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.     

Convolution products of probability measures on completely simple semigroups

Convolution products of probability measures are considered in the context of completely simple semigroups. Given a sequence of measures (μn)⊂Prob(S), where S is a finite completely simple semigroup, results are proven which (1) relate the supports of the measures in the sequence to the supports of the tail limit measures, and (2) determine necessary and sufficient conditions for convergence of the convolution products in the case of rectangular groups. An example showing how the theory can be used to analyze the convergence behavior of non-homogeneous Markov chains is included.     

Almost sure limit theorem for the maximum of a classof quasi-stationary sequences简

This paper investigates the problem of almost sure limit theorem for the maximum of quasi-stationary sequence based on the result of Turkman and Walker. We prove an almost sure limit theorem for the maximum of a class of quasi-stationary sequence under weak dependence conditions of D(uk,un) and αtn,ln = O(log log n).(1+ε).     

Central and local limit theorems for the coefficients of polynomials of binomial type

We introduce the problem of establishing a central limit theorem for the coefficients of a sequence of polynomials P_n(x) of binomial type; that is, a sequence P_n(x) satisfying $\text{exp}\text{(xg(u)) = ∑}{}_{\mathrm{n=0}}{}^{\mathrm{\infty }}\text{P}{}_{\mathrm{n}}\text{(x)(}\text{u}{}^{\mathrm{n}}\text{n!}\text{)}$ for some (formal) power series g(u) lacking constant term. We give a complete answer in the case when g(u) is a polynomial, and point out the widest known class of nonpolynomial power series g(u) for which the corresponding central limit theorem is known true. We also give the least restrictive conditions known for the coefficients of P_n(x) which permit passage from a central to a local limit theorem, as well as a simple criterion for the generating function g(u) which assures these conditions on the coefficients of P_n(x). The latter criterion is a new and general result concerning log concavity of doubly indexed sequences of numbers with combinatorial significance. Asymptotic formulas for the coefficients of P_n(x) are developed.     

On strong limit theorems concerning delayed sums of a random sequence

Let (ξ _n ) _n∈N be a sequence of arbitrarily dependent random variables. In this paper, a generalized strong limit theorem of the delayed average of (ξ _n ) _n∈N is investigated, then some limit theorems for arbitrary information sources follow. As corollaries, some known results are generalized.     

On the divergence of polynomial interpolation

Consider a triangular interpolation scheme on a continuous piecewise C¹ curve of the complex plane, and let Γ be the closure of this triangular scheme. Given a meromorphic function f with no singularities on Γ, we are interested in the region of convergence of the sequence of interpolating polynomials to the function f. In particular, we focus on the case in which Γ is not fully contained in the interior of the region of convergence defined by the standard logarithmic potential. Let us call Γ_out the subset of Γ outside of the convergence region.In the paper we show that the sequence of interpolating polynomials, {P_n}_n, is divergent on all the points of Γ_out, except on a set of zero Lebesgue measure. Moreover, the structure of the set of divergence is also discussed: the subset of values z for which there exists a partial sequence of {P_n(z)}_n that converges to f(z) has zero Hausdorff dimension (so it also has zero Lebesgue measure), while the subset of values for which all the partials are divergent has full Lebesgue measure.The classical Runge example is also considered. In this case we show that, for all z in the part of the interval (−5,5) outside the region of convergence, the sequence {P_n(z)}_n is divergent.     

Reproducing kernel Hilbert spaces and the law of the iterated logarithm for Gaussian processes

In this paper, we find the limit set of a sequence (2 log n)^?1/2 X _n (t), n≧3) of Gaussian processes in C [0,1], where the processes X _n (t) are defined on the same probability space and have the same distribution. Our result generalizes the theorems of Oodaira and Strassen, and we also apply it to obtain limit theorems for stationary Gaussian processes, moving averages of the type \(\int\limits_0^t {f\left( {t - s} \right)dW\left( s \right)} \) , where W(s) is the standard Wiener process, and other Gaussian processes. Using certain properties of the unit ball of the reproducing kernel Hubert space of X _n (t), we derive the usual law of the iterated logarithm for Gaussian processes. The case of multidimensional time is also considered.     

Fluctuations of the empirical quantiles of independent Brownian motions

We consider iid Brownian motions, B_j(t), where B_j(0) has a rapidly decreasing, smooth density function f. The empirical quantiles, or pointwise order statistics, are denoted by B_j:n(t), and we consider a sequence Q_n(t)=B_j(n):n(t), where j(n)/n→α∈(0,1). This sequence converges in probability to q(t), the α-quantile of the law of B_j(t). We first show convergence in law in C[0,∞) of F_n=n^1/2(Q_n−q). We then investigate properties of the limit process F, including its local covariance structure, and Hölder-continuity and variations of its sample paths. In particular, we find that F has the same local properties as fBm with Hurst parameter H=1/4.     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.     

Central and local limit theorems applied to asymptotic enumeration

Let a double sequence a_n(k) ? 0 be given. We prove a simple theorem on generating functions which can be used to establish the asymptotic normality of a_n(k) as a function of k. Next we turn our attention to local limit theorems in order to obtain asymptotic formulas for a_n(k). Applications include constant coefficient recursions, Stirling numbers, and Eulerian numbers.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

Revisits for transient random walk

For a random walk on the integers define R_n as the number of (distinct) states visited in the first n steps and Z_n as the number of states visited in the first n steps which are never revisited. Here we deal with transient walks. The increments of Z_n form a stationary process and various central limit results and an iterated logarithm result are obtained for Z_n from known results on stationary processes. Furthermore, the limit behaviour of R_n is closely related to that of Z_n; this relationship is elucidated and corresponding limit results for R_n are then read off from those for Z_n.     

Central and local limit theorems applied to asymptotic enumeration II: Multivariate generating functions

Let a multivariate sequence a_n(k) ? 0 be given. Multivariate central and local limit theorems are proved for a_n(k) as n → ∞ that are based on examining the generating function. Applications are made to permutations with rises and falls, ordered partitions of sets, Tutte polynomials of recursive families, and dissections of polygons.     

Limit theorems for weighted sums of random elements in separable Banach spaces

Let a_nk, n ≥ 1, k ≥ 1, be a double array of real numbers and let V_n, n ≥ 1, be a sequence of random elements taking values in a separable Banach space. In this paper, we examine under what conditions the sequence Σ_k≥1a_nkV_k, n ≥ 1, has a limit either in probability or almost surely.