Large deviations for sums of i.i.d. random compact sets

We prove a large deviation principle for Minkowski sums of i.i.d. random compact sets in a Banach space, that is, the analog of Cramér theorem for random compact sets.

     

Invariance principles for renewal processes when only moments of low order exist

Starting from recent strong and weak approximations to the partial sums of i.i.d. random vectors (cf. U. Einmahl, Ann. Probab., 15 1419–1440), some corresponding invariance principles are developed for associated renewal processes and random sums. Optimality of the approximation is proved in the case when only two moments exist. Among other applications, a Darling-Erdös type extreme value theorem for renewal processes will be derived.     

Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments

The aim of this paper is to derive consequences of a result of G?tze and Zaitsev (2008). We show that the i.i.d. case of this result implies a multidimensional version of some results of Sakhanenko (1985). We establish bounds for the rate of strong Gaussian approximation of sums of i.i.d. R ^d -valued random vectors ξ _j having finite moments E IIξ _j II^γ, γ>2. Bibliography: 13 titles.     

Estimates for the rate of strong Gaussian approximation for sums of i.i.d. multidimensional random vectors

The aim of this paper is to derive new optimal bounds for the rate of strong Gaussian approximation of sums of i.i.d. R ^d-valued random variables ξ_j that have finite moments of the form EH (‖ξ_j‖), where H (x) is a monotone function growing not slower than x² and not faster than e^cx. We obtain some generalization and improvements of results of U. Einmahl (1989). Bibliography: 28 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 141–158.     

Mixing Limit Theorems of Maximum of Absolute Sums and their Convergence Rates

Conditioned, in the sense of Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228 1958), limit theorem in the L_p-norm of the maximum of absolute sums of independent identically distributed random variables is established and its exact rate of convergence is given. The results are equivalent to establishing analogous results for the supremum of random functions of partial sums defined on C[0,1], i.e., the invariance principle. New methodologies are used to prove the results that are profoundly different from those used in Rényi (Acta Math. Acad. Sci. Hungar. 9, 215–228, 1958) and subsequent authors in proving the conditioned central limit theorem for partial sums.     

Wittmann's Law of Iterated Logarithm for Tail Sums of B-Valued Random Variables

We present an analogue of Wittmann's law of iterated logarithm (LIL) for tail sums of independent B-valued random variables by using the isoperimetric method and give the precise value of the upper limit for the LIL for tail sums.     

Convergence to a stationary state and diffusion for a charged particle in a standing medium

Summary We study a one-dimensional semi-infinite system of identical particles, driven by a constant force acting only on the first particle. Particles interact through elastic collisions. At time zero all particles are at rest, and the interparticle distances are i.i.d. r.v.'s, the support of the distribution being in (d, ), d>0. We show that if d is large enough the dynamics has a strong cluster property, and prove, for large times, convergence to a limiting distribution for the system as seen from the first particle, as well as existence of drift velocity and invariance principle for the motion of the first particle.Partially supported by C.N.R.-C.N.Pq. agreementPartially supported by M.P.I. research funds     

Rosenthal type inequalities forB-valued strong mixing random fields and their applications

Some inequalities for moments of partial sums of aB -valued strong mixing field are established and their applications to the weak and strong laws of large numbers and the complete convergences are discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19701011) and China Postdoctoral Science Foundation.     

An Inequality for Large Deviation Probabilities of Sums of Bounded i.i.d. Random Variables

We provide precise bounds for tail probabilities, say {M _n x}, of sums M _n of bounded i.i.d. random variables. The bounds are expressed through tail probabilities of sums of i.i.d. Bernoulli random variables. In other words, we show that the tails are sub-Bernoullian. Sub-Bernoullian tails are dominated by Gaussian tails. Possible extensions of the methods are discussed.     

Martingale convergence,series expansion of Gaussian elements,and strong law of large numbers in Fréchet spaces

The main result of this paper is the derivation of a convergence theorem for certain martingales with values in a separable Fréchet space F. It is shown that this result includes a well known theorem due to Chatterji. Moreover, the series expansion of zero-mean Gaussian elements with values in F and the strong law of large numbers for i.i.d. F-valued random elements also follow as applications of the main theorem.     

Convergence rates in the law of logarithm of random elements

1. IntroductionLet {Xu, n 2 1} be a sequence of r.v.IS in the same probability space and put Sa =nZ Xi, n 2 1; L(x) = mad (1, logx).i=1Since the definition of complete convergence is illtroduced by Hsu and Robbins[6], therehave been many authors who devote themselves to the study of the complete convergence forsums of i.i.d. real-valued r.v.'s, and obtain a series of elegys results, see [3,7]. Meanwhile,the convergence rates in the law of logarithm of i.i.d. real-vained r.v.'s have also be…     

The Darling-Erdős theorem for sums of I.I.D. random variables

Summary We obtain a general Darling-Erds type theorem for the maximum of appropriately normalized sums of i.i.d. mean zero r.v.'s with finite variances. We infer that the Darling-Erds theorem holds in its classical formulation if and only ifE[X ² 1 {|X|t}]=o((loglogt)^-1) ast. Our method is based on an extension of the truncation techniques of Feller (1946) to non-symmetric r.v.'s. As a by-product we are able to reprove fundamental results of Feller (1946) dealing with lower and upper classes in the Hartman-Wintner LIL.     

A General Darling–Erdős Theorem in Euclidean Space

We provide an improved version of the Darling–Erd?s theorem for sums of i.i.d. random variables with mean zero and finite variance. We extend this result to multidimensional random vectors. Our proof is based on a new strong invariance principle in this setting which has other applications as well such as an integral test refinement of the multidimensional Hartman–Wintner LIL. We also identify a borderline situation where one has weak convergence to a shifted version of the standard limiting distribution in the classical Darling–Erd?s theorem.     

Moments of the maximum of normed partial sums of random variables with multidimensional indices

Summary For a set of i.i.d. random variables indexed by the positive integer d-dimensional lattice points we give conditions for the existence of moments of the supremum of normed partial sums, thereby obtaining results related to the Kolmogorov-Marcinkiewicz strong law of large numbers and the law of the iterated logarithm.     

The Law of the Iterated Logarithm for m-Dependent Banach Space Valued Random Variables

A theorem on the law of the iterated logarithm is established for m-dependent B-valued random variables. The conditions in our theorem match their independent analogues and appear as necessary or minimal for the results. According to an example given in the paper, the situation we face is much different from the finite dimensional case and therefore, so is the way we solve the problems.     

Partial Sum Process for Records

Suppose the upper records from a sequence of i.i.d. random variables is in the domain of attraction of a normal distribution. Consider the D(0,1]-valued process {Z_n(·)} constructed by usual interpolation of the partial sums of the records. We prove that under some mild conditions, {Z_n} converges to a limiting Gaussian process in D(0,1]. As a consequence, the partial sums of records is asymptotically normal. AMS 2000 Subject Classification Primary—60F17 Secondary—60G70     

Laplace approximations for sums of independent random vectors

Summary LetX _i,iN, be i.i.d.B-valued random variables whereB is a real separable Banach space, and a mappingB R. Under some conditions an asymptotic evaluation of is possible, up to a factor (1+o(1)). This also leads to a limit theorem for the appropriately normalized sums under the law transformed by the density exp .     

Logarithmic residues, generalized idempotents, and sums of idempotents in Banach algebras

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.     

Asymptotic expansions forL- andR-statistics under alternatives

In this paper we establish asymptotic expansions (a.e.) under alternatives for the distribution functions of sums of independent identically distributed random variables (i.i.d.r.v.'s.), linear combinations of order statistics, and one-sample rank statistics (L- and R-statistics). The general Lemma from [V. E. Bening,Bull. Moscow State Univ., Ser. 15, 2 36–44 (1994)] is applied to these statistics. Section 1 contains the statement of the theorem, in Sec. 2 the theorems is proved; its proof involves four auxiliary lemmas, also contained in Sec. 2. Finally Sec. 3 contains the proofs of these lemmas. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, 1993.     

COMPLETE CONVERGENCE FOR A CLASS OF MIXING SEQUENCES

Under some conditions on probability, the author obtains some results on the complete convergence for partial sums of not necessary identically distributed ρ-mixing sequences, and the complete convergence for partial sums of B-valued martingale differences is also studied. As application the author gives the corresponding results on the complete convergence for randomly indexed partial sums.