A strong law of large numbers on the harmonic <Emphasis Type="Italic">p</Emphasis>-combination

In this paper, we establish a strong law of large numbers for the harmonic p-combinations of random star bodies. Starting from this theorem, we prove a strong law of large numbers in L _p space and provide the probabilistic version of dual Brunn-Minkowski inequality.     

On the weak laws of large numbers for arrays of random variables

In this paper we obtain weak laws of large numbers (WLLNs) for arrays of random variables under the uniform Cesàro-type condition. As corollary, we obtain the result of Hong and Oh [Hong, D. H., Oh, K. S., 1995. On the weak law of large numbers for arrays. Statist. Probab. Lett. 22, 55–57]. Furthermore, we obtain a WLLN for an L_p-mixingale array without the conditions that the mixingale is uniformly integrable and the L_p-mixingale numbers decay to zero at a special rate.     

Theorems of law of large numbers type for functionals of products of independent random matrices

A limit theorem of the type of the law of large numbers is proved using the martingale method.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 16–19, 1987.     

On the strong law of large numbers for sequences of random variables without the independence condition

New sufficient conditions for the applicability of the strong law of large numbers to a sequence of dependent random variables X ₁, X ₂, …, with finite variances are established. No particular type of dependence between the random variables in the sequence is assumed. The statement of the theorem involves the classical condition Σ _n ^∞ (log₂ n)²/n ² < ∞, which appears in various theorems on the strong law of large numbers for sequences of random variables without the independence condition.     

On the complete convergence of sums of negatively associated random variables

This paper extends results on complete convergence in the law of large numbers for subsequences to the case of negatively associated nonidentically distributed random variables. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 411–420, September, 2000.     

Strong Limit Theorems for Increments of Sums of Independent Random Variables

We derive universal strong laws for increments of sums of independent, nonidentically distributed, random variables. These results generalize universal results of the author for the i.i.d. case which include the strong law of large numbers, law of the iterated logarithm, Erdos-Renyi law, and Csorgo-Revesz laws. Bibliography: 27 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 260–285.     

Limit theorems for sums of partial quotients of continued fractions

We prove that the partial quotientsa _j of the regular continued fraction expansion cannot satisfy a strong law of large numbers for any reasonably growing norming sequence, and that thea _j belong to the domain of normal attraction to a stable law with characteristic exponent 1. We also show that thea _j satisfy a central limit theorem if a few of the largest ones are trimmed.In memory of Wilfried Nöbauer     

Branching structure of uniform recursive trees

The branching structure of uniform recursive trees is investigated in this paper. Using the method of sums for a sequence of independent random variables, the distribution law of ηn, the number of branches of the uniform recursive tree of size n are given first. It is shown that the strong law of large numbers, the central limit theorem and the law of iterated logarithm for ηn follow easily from this method. Next it is shown that ηn and ξn, the depth of vertex n, have the same distribution, and the distribution law of ζn,m, the number of branches of size m, is also given, whose asymptotic distribution is the Poisson distribution with parameter λ= 1/m. In addition, the joint distribution and the asymptotic joint distribution of the numbers of various branches are given. Finally, it is proved that the size of the biggest branch tends to infinity almost sure as n→∞.     

Convergence of Weighted Sums and Laws of Large Numbers in D([0,1]; E)

Convergence properties of weighted sums of functions in D([0, 1]; E) (E a Banach space) are investigated. We show that convergence in the Skorokhod J₁-topology of a sequence (x_n) in D([0, 1]; E) does not imply convergence of a sequence ( _n) of averages. Convergence in the J₁-topology of a sequence ( _n) of averages is shown, under the growth condition x_n ∞ = o(n), to be equivalent to the convergence of ( _n) in the uniform topology. Convergence of a sequence (x_n,) is shown to imply convergence of the sequence ( _n) of averages in the M₁ and M₂ topologies. The strong law of large numbers in D[0, 1] is considered and an example is constructed to show that different definitions of the strong law of large numbers are nonequivalent.     

Convergence of weighted sums of random functions in D[0, 1]

Conditions are investigated which imply the tightness of certain weighted sums Σ_{i = 1}^k_n a_niX_i of random functions (X_n) taking values in D([0, 1]; E), where E is a separable Banach space. Improved weak laws of large numbers result as corollaries. Examples are presented to clarify the relative strengths of the moment conditions and their relationship to tightness and the strong law of large numbers. A tightness condition is defined using a certain class of sets measurable in the Skorokhod J₁-topology, which yields J₁-tightness of sequences of weighted sums. As a consequence, tightness of a sequence (X_n) in the Skorokhod M₁-topology is used to obtain J₁-tightness of a sequence ( ) of averages and a strong law of large numbers in D(R₊).     

Estimation of distribution tails for normalized and self-normalized sums

A combinatorial inequality is derived. This inequality is applied to obtain new estimates for probabilities of large deviations of normalized and self-normalized sums of independent and dependent positive random values. As a consequence, an estimate from above is derived for the strong law of large numbers. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 77–87.This research was supported in part by the Ministry of Education of Russia, grant E00-1.0-45, and by the Russian Foundation for Basic Research, grant 02-01-01099a.Translated by V. A. Egorov.     

Strong Laws of Large Numbers by Elementary Tauberian Arguments

For Kolmogorovs strong law of large numbers an alternative short proof is given which weakens Etemadis condition of pairwise independence. The argument uses the known – and elementary – equivalence of (Cesàro) C₁- and C₂-summability for one-sided bounded sequences. Also other strong laws of large numbers are established, in part via Borel summability.     

Strong Laws of Large Numbers by Elementary Tauberian Arguments

For Kolmogorovs strong law of large numbers an alternative short proof is given which weakens Etemadis condition of pairwise independence. The argument uses the known – and elementary – equivalence of (Cesàro) C₁- and C₂-summability for one-sided bounded sequences. Also other strong laws of large numbers are established, in part via Borel summability.     

Limit theorems for numbers of near-records

Observations occurring between successive record times and within a distance a > 0 of the current record value are called near-records. Limit theorems for the number ξ _n (a) of near records are found for cases in which the parent distribution lies in a maximal domain of attraction and a is a function of n. Corollaries are indicated for numbers of near-k-records and sums of near-records. If the parent law is thin-tailed and a is constant, then a centered and normed version of logξ _n (a) has a limit law under appropriate conditions.      

A Random Set Extension of a Strong Law of Large Numbers of Cantrell and Rosalsky

Abstract

A strong law of large numbers under conditions irrespective of the joint distribution of the sequence is extended to random sets. The extension is such that the role of events of the form {||V _n || ≤ b _n } (where V _n is a random element of a separable Banach space) is played by events of the form {X _n  ? B _n } (where X _n is a random closed bounded set).     

Marcinkiewicz–Zygmund Type Strong Law of Large Numbers for Pairwise i.i.d. Random Variables

Etemadi (in Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119–122, 1981) proved that the Kolmogorov strong law of large numbers holds for pairwise independent identically distributed (pairwise i.i.d.) random variables. However, it is not known yet whether the Marcinkiewicz–Zygmund strong law of large numbers holds for pairwise i.i.d. random variables. In this paper, we obtain the Marcinkiewicz–Zygmund type strong law of large numbers for pairwise i.i.d. random variables {X _n ,n≥1} under the moment condition E|X ₁| ^p (loglog|X ₁|)^2(p?1)<∞, where 1<p<2.     

Asymptotical behavior of the variance of sums of random variables with random replacements

In this paper, we introduce a scheme of summation of independent random variables with random replacements. We consider a series of double arrays of identically distributed random variables that are row-wise independent, but such that neighboring rows contain a random common part of the repeating terms. By this-scheme we bscribe a model of strongly dependent noise. To investigate the sample mean of this noise, we consider the sum of random variables over the whole double array and its conditional variance with respect to replacements. For columns of the arrays we prove a covariance inequality. As a corollary of it, we demonstrate the law of large numbers for conditional variances. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 124–143. Translated by A. Sudakov.     

Inverse optimal control and construction of control Lyapunov functions

In this paper, the construction of CLFs for nonlinear systems and a new inverse optimal control law are presented. The construction of a CLF for an affine nonlinear system is reduced to the construction of a CLF for a simpler system, and a new L _g V type control law with respect to a CLF is provided. This control law is a generalization of Sontag’s formula and contains a design parameter. Tuning this parameter gives many suboptimal solutions for the optimization problem. Also, the gain margin and sector margin of the control law are calculated. Examples are provided to illustrate the main theoretical results of the paper. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.     

Measures on diffeomorphism groups for non-archimedean manifolds: Group representations and their applications

Nondegenerate σ-additive measures with ranges in ℝ and ℚ_q (q≠p are prime numbers) that are quasi-invariant and pseudodifferentiable with respect to dense subgroups G′ are constructed on diffeomorphism and homeomorphism groups G for separable non-Archimedean Banach manifolds M over a local fieldK,K ⊃ ℚ_q, where ℚ_q is the field of p-adic numbers. These measures and the associated irreducible representations are used in the non-Archimedean gravitation theory. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 3, pp. 381–396, June, 1999.     

A class of bounds for convex bodies in Hilbert space

We extend to infinite dimensions a class of bounds forL _p metrics of finite-dimensional convex bodies. A generalization to arbitrary increasing convex functions is done simultaneously. The main tool is the use of Gaussian measure to effect a normalization for varying dimension. At a point in the proof we also invoke a strong law of large numbers for random sets to produce a rotational averaging.Supported in part by ONR Grant N0014-90-J-1641 and NSF Grant DMS-9002665.