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.     

Discrete random bounds for general random variables and applications to reliability

We here propose some new algorithms to compute bounds for (1) cumulative density functions of sums of i.i.d. nonnegative random variables, (2) renewal functions and (3) cumulative density functions of geometric sums of i.i.d. nonnegative random variables. The idea is very basic and consists in bounding any general nonnegative random variable X   by two discrete random variables with range in hN$h\mathbb{N}$, which both converge to X as h goes to 0. Numerical experiments are lead on and the results given by the different algorithms are compared to theoretical results in case of i.i.d. exponentially distributed random variables and to other numerical methods in other cases.     

Loi du logarithme itéré pour les composantes du modèle additif de régression

In the setting of the additive model of the regression function, we study the iterated logarithm law for this model components pertaining with the marginal integration estimation method. Our results are stated in the i.i.d. random vectors framework. To cite this article: M. Debbarh, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Implicit Sampling, with Application to Data Assimilation

There are many computational tasks, in which it is necessary to sample a given probability density function (or pdf for short), i.e., to use a computer to construct a sequence of independent random vectors x i (i = 1, 2, ··· ), whose histogram converges to the given pdf. This can be difficult because the sample space can be huge, and more importantly, because the portion of the space, where the density is significant, can be very small, so that one may miss it by an ill-designed sampling scheme. Indeed, Markovchain Monte Carlo, the most widely used sampling scheme, can be thought of as a search algorithm, where one starts at an arbitrary point and one advances step-by-step towards the high probability region of the space. This can be expensive, in particular because one is typically interested in independent samples, while the chain has a memory. The authors present an alternative, in which samples are found by solving an algebraic equation with a random right-hand side rather than by following a chain; each sample is independent of the previous samples. The construction in the context of numerical integration is explained, and then it is applied to data assimilation.     

On complete convergence and the strong law of large numbers for pairwise independent random variables

The complete convergence for the maximum partial sums of pairwise independent random variables is obtained. The Kolmogorov strong law of large numbers for pairwise i.i.d. random variables is also obtained. Similar results are established for the moving average processes of pairwise i.i.d. random variables and for pairwise independent random elements taking values in a Banach space.     

Analysis of R out of N systems with several repairmen,exponential life times and phase type repair times: An algorithmic approach

An R out of N repairable system consisting of N independent components is operating if at least R components are functioning. The system fails whenever the number of good components decreases from R to R − 1. A failed component is sent to a repair facility having several repairmen. Life times of working components are i.i.d random variables having an exponential distribution. Repair times are i.i.d random variables having a phase type distribution. Both cold and warm stand-by systems are considered. We present an algorithm deriving recursively in the number of repairmen the generator of the Markov process that governs the process. Then we derive formulas for the point availability, the limiting availability, the distribution of the down time and the up time. Numerical examples are given for various repair time distributions. The numerical examples show that the availability is not very sensitive to the repair time distribution while the mean up time and the mean down time might be very sensitive to the repair time distributions.     

Maximum likelihood estimation in the context of a sub-ballistic random walk in a parametric random environment

We consider a one-dimensional sub-ballistic random walk evolving in a parametric i.i.d. random environment. We study the asymptotic properties of the maximum likelihood estimator (MLE) of the parameter based on a single observation of the path till the time it reaches a distant site. For that purpose, we adapt the method developed in the ballistic case by Comets et al. (2014) and Falconnet et al. (2014). Using a supplementary assumption due to the special nature of the sub-ballistic regime, we prove consistency and asymptotic normality as the distant site tends to infinity. To emphasize the role of the additional assumption, we investigate the Temkin model with unknown support, and it turns out that the MLE is consistent but, unlike the ballistic regime, the Fisher information is infinite. We also explore the numerical performance of our estimation procedure.     

Size of random Galois lattices and number of closed frequent itemsets

Given a sample of binary random vectors with i.i.d. Bernoulli(p) components, that is equal to 1 (resp. 0) with probability p (resp. 1−p), we first establish a formula for the mean of the size of the random Galois lattice built from this sample, and a more complex one for its variance. Then, noticing that closed α-frequent itemsets are in bijection with closed α-winning coalitions, we establish similar formulas for the mean and the variance of the number of closed α-frequent itemsets. This can be interesting for the study of the complexity of some data mining problems such as association rule mining, sequential pattern mining and classification.     

On χ2-test of goodness-of-fit for a class of discrete multivariate models

A model for an n-dimensional random vector which is the sum of v i.i.d. random variables with an exponential family distribution is considered. It is shown that the exact test can be approximated by a χ²-test as v → ∞. As an example a goodness-of-fit test for Poisson-distribution based on observed frequencies is derived.     

Quenched invariance principle for random walks in balanced random environment

We consider random walks in a balanced random environment in ${\mathbb{Z}^d}$ , d?≥ 2. We first prove an invariance principle (for d?≥ 2) and the transience of the random walks when d?≥ 3 (recurrence when d?=?2) in an ergodic environment which is not uniformly elliptic but satisfies certain moment condition. Then, using percolation arguments, we show that under mere ellipticity, the above results hold for random walks in i.i.d. balanced environments.     

Managing stochastic inventory systems with free shipping option

In many industries, customers are offered free shipping whenever an order placed exceeds a minimum quantity specified by suppliers. This allows the suppliers to achieve economies of scale in terms of production and distribution by encouraging customers to place large orders. In this paper, we consider the optimal policy of a retailer who operates a single-product inventory system under periodic review. The ordering cost of the retailer is a linear function of the ordering quantity, and the shipping cost is a fixed constant K whenever the order size is less than a given quantity – the free shipping quantity (FSQ), and it is zero whenever the order size is at least as much as the FSQ. Demands in different time periods are i.i.d. random variables. We provide the optimal inventory control policy and characterize its structural properties for the single-period model. For multi-period inventory systems, we propose and analyze a heuristic policy that has a simple structure, the (s, t, S) policy. Optimal parameters of the proposed heuristic policy are then computed. Through an extensive numerical study, we demonstrate that the heuristic policy is sufficiently accurate and close to optimal.     

Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability

From the classical notion of uniform integrability of a sequence of random variables, a new concept of integrability (called h-integrability) is introduced for an array of random variables, concerning an array of constants. We prove that this concept is weaker than other previous related notions of integrability, such as Cesàro uniform integrability [Chandra, Sankhyā Ser. A 51 (1989) 309-317], uniform integrability concerning the weights [Ordóñez Cabrera, Collect. Math. 45 (1994) 121-132] and Cesàro α-integrability [Chandra and Goswami, J. Theoret. Probab. 16 (2003) 655-669].Under this condition of integrability and appropriate conditions on the array of weights, mean convergence theorems and weak laws of large numbers for weighted sums of an array of random variables are obtained when the random variables are subject to some special kinds of dependence: (a) rowwise pairwise negative dependence, (b) rowwise pairwise non-positive correlation, (c) when the sequence of random variables in every row is φ-mixing. Finally, we consider the general weak law of large numbers in the sense of Gut [Statist. Probab. Lett. 14 (1992) 49-52] under this new condition of integrability for a Banach space setting.     

Procedure for determining the dispersion and correlation matrices for a multidimensional random process

A method is proposed for determining the dispersion and correlation matrices for a random process that can be represented as a system of differential equations. The method provides a significant reduction of computational operations and is free of the systematic error that accompanies numerical integration. As an example we present finite expressions for the probability characteristics of observed variables.Translated fromVychislitel'naya i Prikladnaya Matematika, No. 69, pp. 135–141, 1989.     

Numerical integration in Galerkin meshless methods, applied to elliptic Neumann problem with non-constant coefficients

In this paper, we explore the effect of numerical integration on the Galerkin meshless method used to approximate the solution of an elliptic partial differential equation with non-constant coefficients with Neumann boundary conditions. We considered Galerkin meshless methods with shape functions that reproduce polynomials of degree k?≥?1. We have obtained an estimate for the energy norm of the error in the approximate solution under the presence of numerical integration. This result has been established under the assumption that the numerical integration rule satisfies a certain discrete Green’s formula, which is not problem dependent, i.e., does not depend on the non-constant coefficients of the problem. We have also derived numerical integration rules satisfying the discrete Green’s formula.     

ɛ-nets and simplex range queries

We demonstrate the existence of data structures for half-space and simplex range queries on finite point sets ind-dimensional space,d≥2, with linear storage andO(n ^α ) query time, $$\alpha = \frac{{d(d - 1)}}{{d(d - 1) + 1}} + \gamma for all \gamma > 0$$ . These bounds are better than those previously published for alld≥2. Based on ideas due to Vapnik and Chervonenkis, we introduce the concept of an ?-net of a set of points for an abstract set of ranges and give sufficient conditions that a random sample is an ?-net with any desired probability. Using these results, we demonstrate how random samples can be used to build a partition-tree structure that achieves the above query time.     

Crossing velocities for an annealed random walk in a random potential

We consider a random walk in an i.i.d. non-negative potential on the d-dimensional integer lattice. The walk starts at the origin and is conditioned to hit a remote location y on the lattice. We prove that the expected time under the annealed path measure needed by the random walk to reach y grows only linearly in the distance from y to the origin. In dimension 1 we show the existence of the asymptotic positive speed.     

Simulation de lois conjuguées et application à l'estimation de la transformée de Cramer dans ℝd

Let X, X₁, X₂,… be a sequence of i.i.d. ℝ^d-valued random variables with distribution F. An algorithm for the simulation of random vectors with distribution dF_t (x):= e〈t,x〉dF(x)/(t), where (t):= Ee^ť,X〉 (cf. [2]) is used for the estimation of the Cramer transform H (x):= sup_t (〈t, X〉 − log(t)). This method, which belongs to the class of “acceptance-rejection” techniques, is fast and uses a random sieve on the sequence (X_i)_{i ≥ 1}; it does not assume any prior knowledge on F or . We state the asymptotic properties of this estimator calculated on a n-sample of simulated r.v. 's with distribution F_t. We also present some numerical simulations.     

Point Sets on the Sphere \mathbb{S}^{2} with Small Spherical Cap Discrepancy

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order?N ^?1/2. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A?detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given.     

A dyadic age-replacement policy for a periodically inspected equipment item subject to random deterioration

A periodic review replacement system is considered. The amount of deterioration over successive periods forms a sequence of i.i.d. random variables. A replacement policy of the dyadic type is in effect whereby the used equipment item is discarded and immediately replaced by a new identical equipment item if at the end of a period the old equipment has service aged by an amount in excess of S or has been in operation for exactly N periods whichever comes first. Using a theorem on renewal reward processes, an expression for the total steady-state expected cost per period is derived, consisting of a fixed replacement cost and a linear cost of operation. Optimal values of S and N that minimize this steady state cost are computed for a few numerical examples, when the service aging per period has a gamma distribution.     

The Littlewood-Offord problem and invertibility of random matrices

We prove two basic conjectures on the distribution of the smallest singular value of random n×n matrices with independent entries. Under minimal moment assumptions, we show that the smallest singular value is of order n−1/2, which is optimal for Gaussian matrices. Moreover, we give a optimal estimate on the tail probability. This comes as a consequence of a new and essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random variables Xk and real numbers ak, determine the probability p that the sum k_∑akXk lies near some number v. For arbitrary coefficients ak of the same order of magnitude, we show that they essentially lie in an arithmetic progression of length 1/p.