Marcinkiewicz Laws Research Articles

Strong Limit Theorems for Dependent Random Variables

In this article We establish moment inequality of dependent random variables, furthermore some theorems of strong law of large numbers and complete convergence for sequences of dependent random variables. In particular, independent and identically distributed Marcinkiewicz Law of large numbers are generalized to the case of m₀ -dependent sequences.

Analysis of the finiteness for the first collision time between two randomly moving particles

The finiteness of the collision time between two different randomly moving particles is presented by providing more useful analysis that gives stronger and finite moment. The triangular arrays and the uniform integrability conditions of the all probable positions non-stationary random sequence are used. In addition, an important property of Marcinkiewicz laws of large numbers and Hoffman-Jorgensen inequality are presented in this analysis. All of them are deriving to provide the sufficient conditions that give more stronger moments of the first meeting time in the probability space.

Marcinkiewicz law of large numbers for outer products of heavy-tailed, long-range dependent data

Abstract The Marcinkiewicz strong law, limn→∞(1 / n1/p)∑k=1n(Dk - D) = 0 almost surely with p ∈ (1, 2), is studied for outer products Dk = {XkX̅kT}, where {Xk} and {X̅k} are both two-sided (multivariate) linear processes (with coefficient matrices (Cl), (C̅l) and independent and identically distributed zero-mean innovations {Ξ} and {Ξ̅}). Matrix sequences Cl and C ̅l can decay slowly enough (as |l| → ∞) that {Xk,X ̅k} have long-range dependence, while {Dk} can have heavy tails. In particular, the heavy-tail and long-range-dependence phenomena for {Dk} are handled simultaneously and a new decoupling property is proved that shows the convergence rate is determined by the worst of the heavy tails or the long-range dependence, but not the combination. The main result is applied to obtain a Marcinkiewicz strong law of large numbers for stochastic approximation, nonlinear function forms, and autocovariances.

THE CONVERGENCE RATES IN THE ASYMMETRIC LAWS OF LARGE NUMBER FOR NEGATIVELY ASSOCIATED RANDOM FIELDS

Convergence rates in the law of large numbers for i.i.d. random variables have been generalized by Gut[Gut, A., 1978. Marc inkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab. 6, 469-482] to random fields with all indices having the same power in the normalization. In this paper we generalize these convergence rates to the identically distributed and negatively associated random fields with different indices having different power in the normalization.

On the Hsu–Robbins–Erdős–Spitzer–Baum–Katz theorem for random fields

The by now classical results on convergence rates in the law of large numbers involving the sums ∑ n = 1 ∞ n α r − 2 P ( | S n | > n α ε ) , where r > 0 , α > 1 / 2 , such that α r ⩾ 1 has been extended to the case α = 1 / 2 by adding additional logarithms. All of this has been generalized to random fields by the first named author in [A. Gut, Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices, Ann. Probab. 6 (1978) 469–482; A. Gut, Convergence rates for probabilities of moderate deviations for sums of random variables with multidimensional indices, Ann. Probab. 8 (1980) 298–313]. The purpose of the present paper is to treat the case when the αʼs differ in the different directions of the field, as well as mixed cases with some αʼs equal to 1/2 with added logarithms and/or iterated ones.

A note on Marcinkiewicz laws for strictly stationary [formula omitted]-mixing sequences

A connection between boundedness in probability and laws of large numbers is established for φ -mixing strictly stationary sequences.

Strong Limit Theorems for Dependent Random Variables

In This Article We establish moment inequality of dependent random variables,furthermore some theorems of strong law of large numbers and complete convergence for sequences of dependent random variables. In particular, independent and identically distributed Marcinkiewicz Law of large numbers are generalized to the case of m0-dependent sequences. Keywords—Lacunary System, Generalized Gaussian, NA sequences, strong law of large numbers.

Marcinkiewicz laws with infinite moments

Marcinkiewicz laws of large numbers for φ-mixing strictly stationary sequences with r-th moment barely divergent, 0 < r < 2, are established. For this dependent analogs of the Levy-Ottaviani-Etemadi and Hoffmann-Jorgensen inequalities are revisited.

Marcinkiewicz laws with infinite moments

Marcinkiewicz laws with infinite moments

An asymmetric Marcinkiewicz–Zygmund LLN for random fields

The classical Marcinkiewicz–Zygmund law for i.i.d. random variables has been generalized by Gut [Gut, A., 1978. Marcinkiewicz laws and convergence rates in the law of large numbers for random variables with multidimensional indices. Ann. Probab. 6, 469–482] to random fields. Therein all indices have the same power in the normalization. Looking into some weighted means of random fields, such as Cesàro summation, it is of interest to generalize these laws to the case where different indices have different powers in the normalization. In this paper we give precise moment conditions for such laws.

The law of the large numbers for U-statistics for 2 m-wise independent random variables

The strong law of the large numbers for U-statistics has been proved for a sequence of independent random variables using martingale techniques. We see that the condition of independence can be relaxed to 2 m-wise independence. In this case martingale techniques cannot be applied. We also consider Marcinkiewicz laws of large numbers for U-statistics for a special case.

The law of the iterated logarithm and Marcinkiewicz law of large numbers forB-valuedU-statistics

Suppose thatB is a separable Banach space and (S,l,P) a probability space.H is a measurable symmetric kernel function fromSm intoB. In this paper we shall further study some limit theorems forB-valuedU-statisticsUmnH based onP andH. Special attention is paid upon the Marcinkiewicz type law of large numbers and the law of the iterated logarithm. Our results can be regarded as extensions of corresponding results for sums of independentB-valued random variables toU-statistics.

Almost sure convergence theorems of weighted sums of random variables

Let be random variables and a triangular array of constants. In this paper we find the various conditions on {ani } and {Xn } under which converges to zero almost surely. It is shown that for the identically distributed random variables the conditions a.s. A generalization of Marcinkiewicz's law of large numbers is given. It is shown that for the i.i.d. case the conditions and EX = 0 imply that a.s

Marcinkiewicz Laws and Convergence Rates in the Law of Large Numbers for Random Variables with Multidimensional Indices

Consider a set of independent identically distributed random variables indexed by $Z^d_+$, the positive integer $d$-dimensional lattice points, $d \geqq 2$. The classical Kolmogorov-Marcinkiewicz strong law of large numbers is generalized to this case. Also, convergence rates in the law of large numbers are derived, i.e., the rate of convergence to zero of, for example, the tail probabilities of the sample sums is determined.