Sequence Of Non-negative Random Variables Research Articles

Estimating function method for nonnegative autoregressive models

A stationary sequence of nonnegative random variables generated by autoregressive (AR) models may be used to describe the inter‐arrival times between events in counting processes. Even though, several such models are available in the literature, there is no unified approach to estimate their parameters. In this paper, we propose a class of combined estimating function method to estimate the model parameters of AR models with gamma marginals. The proposed method is compared with other estimation procedures and are illustrated by simulation and data analysis.

A note on the von Weizsäcker theorem

The von Weizsäcker theorem states that every sequence of nonnegative random variables has a subsequence which is Cesàro convergent to a nonnegative random variable which might be infinite. The goal of this note is to provide a description of the set where the limit is finite. For this purpose, we use a decomposition result due to Brannath and Schachermayer.

On the Rate of Convergence in the Strong Law of Large Numbers for Nonnegative Random Variables

The rate of convergence in the strong law of large numbers for sequences of nonnegative random variables is studied without the independence assumption. Conditions for which an analog of the Baum–Katz theorem holds are obtained.

A strong law of large numbers for nonnegative random variables and applications

For a sequence of nonnegative random variables {Xn,n≥1} with finite means and partial sums Sn=∑i=1nXi,n≥1, and a sequence of positive numbers {bn,n≥1} with bn↑∞, sufficient conditions are given under which (Sn−ESn)/bn→0 almost surely. Our result generalizes the strong law of large numbers obtained by Korchevsky (2015). Some applications for dependent random variables are also provided.

Uniform estimate for the tail probabilities of randomly weighted sums

Several authors have studied the uniform estimate for the tail probabilities of randomly weighted sums \(\sum\limits_{i = 1}^n {\theta _i X_i }\) and their maxima. In this paper, we generalize their work to the situation that {Xi, i ≥ 1} is a sequence of upper tail asymptotically independent random variables with common distribution from the class \(\mathcal{D} \cap \mathcal{L}\), and {θi, i ≥ 1} is a sequence of nonnegative random variables, independent of {Xi, i ≥ 1} and satisfying some regular conditions. Moreover, no additional assumption is required on the dependence structure of {θi, i ≥ 1}.

A theorem on the strong law of large numbers for a sequence of nonnegative random variables

New sufficient conditions are found for the applicability of the strong law of large numbers to a sequence of dependent nonnegative random variables. Bibliography: 3 titles.

Forward-convex convergence in probability of sequences of nonnegative random variables

For a sequence $ (f_n)_{n \in \mathbb{N}}$ of nonnegative random variables, we provide simple necessary and sufficient conditions for convergence in probability of each sequence $ (h_n)_{n \in \mathbb{N}}$ with $ h_n\in \mathrm {conv}(\{f_n,f_{n+1},\dots \})$ for all $ n \in \mathbb{N}$ to the same limit. These conditions correspond to an essentially measure-free version of the notion of uniform integrability.

Fixed points of inhomogeneous smoothing transforms

We consider the inhomogeneous version of the fixed-point equation of the smoothing transformation, that is, the equation , where means equality in distribution, is a given sequence of non-negative random variables and is a sequence of i.i.d. copies of the non-negative random variable X independent of . In this situation, X (or, more precisely, the distribution of X) is said to be a fixed point of the (inhomogeneous) smoothing transform. In the present paper, we give a necessary and sufficient condition for the existence of a fixed point. Furthermore, we establish an explicit one-to-one correspondence with the solutions to the corresponding homogeneous equation with C = 0. Using this correspondence and the known theory on the homogeneous equation, we present a full characterization of the set of fixed points under mild assumptions.

On the strong law of large numbers for a sequence of nonnegative random variables

New sufficient conditions are found for the applicability of the strong law of large numbers to a sequence of dependent nonnegative random variables with finite variances. Bibliography: 4 titles.

Erratum to: Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

This paper investigates the asymptotic behavior of tail probability of randomly weighted sums of dependent and real-valued random variables with dominated variation, where the weights form another sequence of nonnegative random variables. The result we obtain extends the corresponding result of Wang and Tang[7].

Some Intrinsic Properties of the Gamma Distribution

Let {Yn} be a sequence of nonnegative random variables (rvs), and Sn = ∑n j=1 Yj , n ≥ 1. It is first shown that independence of Sk−1 and Yk, for all 2 ≤ k ≤ n, does not imply the independence of Y1, Y2, . . . , Yn. When Yj ’s are identically distributed exponential Exp(α) variables, we show that the independence of Sk−1 and Yk, 2 ≤ k ≤ n, implies that the Sk follows a gamma G(α, k) distribution for every 1 ≤ k ≤ n. It is shown by a counterexample that the converse is not true. We show that if X is a non-negative integer valued rv, then there exists, under certain conditions, a rv Y ≥ 0 such that N(Y ) L = X, where {N(t)} is a standard (homogeneous) Poisson process, and obtain the Laplace-Stieltjes transform of Y . This leads to a new characterization for the gamma distribution. It is also shown that a G(α, k) distribution may arise as the distribution of Sk, where the components are not necessarily exponential. Several typical examples are discussed.

On the Strong Law of Large Numbers for Nonnegative Random Variables

The wide sufficient conditions of applying the strong law of large numbers to a sequence of nonnegative random variables with finite variations are found.

Stochastic modelling of risk control operations for recovery of systems

The paper uses a discrete random variable and two sequences of nonnegative random variables formulating a stochastic model which is the maximum of a random number of stochastic multiplicative models. Sufficient conditions for the distribution function of the model are established. The paper also provides applications of this model assessing certain risk control operations applied to a random number of systems which have been affected by a major risk. Moreover, the paper makes clear that a concept of extreme value theory can contribute to the improvement of the applicability of risk management in a class of systems.

Approximation of the tail probability of randomly weighted sums and applications

Consider the problem of approximating the tail probability of randomly weighted sums ∑ i = 1 n Θ i X i and their maxima, where { X i , i ≥ 1 } is a sequence of identically distributed but not necessarily independent random variables from the extended regular variation class, and { Θ i , i ≥ 1 } is a sequence of nonnegative random variables, independent of { X i , i ≥ 1 } and satisfying certain moment conditions. Under the assumption that { X i , i ≥ 1 } has no bivariate upper tail dependence along with some other mild conditions, this paper establishes the following asymptotic relations: Pr ( max 1 ≤ k ≤ n ∑ i = 1 k Θ i X i > x ) ∼ Pr ( ∑ i = 1 n Θ i X i > x ) ∼ ∑ i = 1 n Pr ( Θ i X i > x ) , and Pr ( max 1 ≤ k < ∞ ∑ i = 1 k Θ i X i > x ) ∼ Pr ( ∑ i = 1 ∞ Θ i X i + > x ) ∼ ∑ i = 1 ∞ Pr ( Θ i X i > x ) , as x → ∞ . In doing so, no assumption is made on the dependence structure of the sequence { Θ i , i ≥ 1 } .

Weak convergence of mutually independent XnBand XnAunder weak convergence of Xn≡XnB-XnA

For each $n\geq 1,$ let $\{ v_{n,k}, k\geq 1\} $ and $\{ u_{n,k}, k\geq 1\} $ be mutually independent sequences of nonnegative random variables and let each of them consist of mutually independent and identically distributed random variables with means $

Large deviations viewpoints for a heavy-tailed β-mixing sequence

Let xnn ⩾ 1 be a stationary sequence of non-negative random variables with heavy tails. Under mixing conditions, we study logarithmic asymptotics for the distributions of the partial sumsSn = X1+ X2 + ... +Xn. We obtain the crude estimatesP(Sn <nx)≈n??x+1 for appropriate values ofx, where ? is a specific parameter. The related conjecture proposed by Gantert is investigated. As a by-product, the so-called supremum large deviations principle is also studied.

Stochastic comparison of multivariate random sums

We establish preservation results for the stochastic comparison of multivariate random sums of stationary, not necessarily independent, sequences of nonnegative random variables. We consider convex-type orderings, i.e. convex, coordinatewise convex, upper

Modeling Sequences of Long Memory Non-negative Covariance Stationary Random Variables

This paper extends the class of covariance stationary GARCH processes of Engle (1982) and Bollerslev (1986) to the case of non-summable autocovariances. We improve on the results of two previous studies in this field: FIGARCH model of Baillie, Bollerslev and Mikkelsen (1996), which generates sequences of non-negative random variables with infinite first and higher-order moments, and hyperbolic decay rate of the impulse response function, and linear ARCH model of Giraitis, Robinson and Surgailis (2000), which does not nest the class of short-memory GARCH processes. We use infinite series representation of GARCH models in terms of martingale differences innovations referred to as MD-ARCH(infinity) representation. This allows for the case of hyperbolically decaying square-summable weighting coefficients. Conditions for non-negativity and covariance stationarity of MD-ARCH(infinity) sequences are derived, and functional limit of the normalized partial sums of the process is studied. Applications of long-memory MD-ARCH(infinity) processes include volatility modeling and high-frequency financial data econometrics.

A uniform limit theorem for predictive distributions

Let { F n} be a filtration, { X n } an adapted sequence of real random variables, and { α n } a predictable sequence of non-negative random variables with α 1>0. Set β n= ∑ i=1 n α i and define the random distribution functions F n(t)=(1/β n) ∑ i=1 n α iI {X i⩽t} and B n(t)=(1/β n) ∑ i=1 n α iP(X i⩽t| F i−1) . Under mild assumptions on { α n }, it is shown that sup t |F n(t)−B n(t)|→0 , a.s. on the set { F n or B n converges uniformly } . Moreover, conditions are given under which F n converges uniformly with probability 1.

On inverse moments of nonnegative random variables

We give sufficient conditions under which E (1+X n) −α ≈(1+ E X n) −α, for a sequence of nonnegative random variables and α>0.