A note on a strong renewal theorem

Let {S_n, n ≥ 1} be a sequence of partial sums of independent and identically distributed non-negative random variables with a common distribution function F. Let F belong to the domain of attraction of a stable law with exponent α, 0 < α < 1. Suppose H(t) = ? N(t), t ? 0, where N(t) = max(n : S_n ≥ t). Under some additional assumptions on F, the difference between H(t) and its asymptotic value as t → ∞ is estimated.     

On almost sure convergence for weighted sums of pairwise negatively quadrant dependent random variables

Let {X _n , n ≥ 1} be a sequence of pairwise negatively quadrant dependent (NQD) random variables. In this study, we prove almost sure limit theorems for weighted sums of the random variables. From these results, we obtain a version of the Glivenko–Cantelli lemma for pairwise NQD random variables under some fragile conditions. Moreover, a simulation study is done to compare the convergence rates with those of Azarnoosh (Pak J Statist 19(1):15–23, 2003) and Li et al. (Bull Inst Math 1:281–305, 2006).     

On the number of near-records after the maximum observation in a continuous sample

{X_n, n≥1} are independent and identically distributed random variables with continuous distribution function F(x). For j=1,…,n, X_j is called a near-record up to time n if X_j ∈ (M_na, M_n], where M_n = max_1≤j≤n {X_j} and a is a positive constant. Let Z_n(a) denote the number of near-records after, and including the maximum observation of the sequence. In this paper, the distributional results of Z_n(a) are considered and its asymptotic behaviours are studied.     

On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

Characterizations of the exponential distribution by some properties of the record values

A sequence {X_n, n≥1} of independent and identically distributed random variables with continuous cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max {X₁, X₂, ..., X_j?1}. We define L(n)=min {j|j>L(n?1), X_j>X_L(n?1)}, with L(0)=1. Let Z_n,m=X_L(n)?X_L(m), n>m≥0. Some characterizations of the exponential distribution are considered in terms of Z_n,m and X_L(m).     

Record values of exponentially distributed random variables

A sequence {X_n, n≥1} of independent and identically distributed random variables with absolutely continuous (with respect to Lebesque measure) cumulative distribution function F(x) is considered. X_j is a record value of this sequence if X_j>max(X₁,…,X_j?1), j>1. Let {X_L(n), n≥0} with L(o)=1 be the sequence of such record values and Z_n,n?1=X_L(n)–X_L(n?1). Some properties of Z_n,n?1 are studied and characterizations of the exponential distribution are discussed in terms of the expectation and the hazard rate of z_n,n?1.     

Strict stationarity of ar(p) processes generated by nonlinear random functions with additive perturbations

Let {X_n} be a generalized autoregressive process of order ρ defined by X_n=Φ_n(X_n-ρ,…,X_n-1)-η_m, where {φ_n} is a sequence of i.i.d. random maps taking values on H, and {η_n} is a sequence of i.i.d. random variables. Let H be a collection of Borel measurable functions on R_P to R. By considering the associated Markov process, we obtain sufficient conditions for stationarity, (geometric) ergodicity of {X_n}.     

Convergence properties for weighted sums of NSD random variables

Abstract

Let {X_n, n ? 1} be a sequence of negatively superadditive dependent (NSD, in short) random variables and {b_ni, 1 ? i ? n, n ? 1} be an array of real numbers. In this article, we study the strong law of large numbers for the weighted sums ∑ⁿ_{i = 1}b_niX_i without identical distribution. We present some sufficient conditions to prove the strong law of large numbers. As an application, the Marcinkiewicz-Zygmund strong law of large numbers for NSD random variables is obtained. In addition, the complete convergence for the weighted sums of NSD random variables is established. Our results generalize and improve some corresponding ones for independent random variables and negatively associated random variables.     

Almost Sure Convergence of Weighted Sums for Negatively Associated Random Variables

In this article, let {X₁, …, X_n} be a sequence of negatively associated random variables and {a_ni, 1 ? i ? n, n ? 1} be a triangular array of constants. Several almost sure convergence theorems for the weighted sums ∑ⁿ_{i = 1}a_niX_i are established.     

Lp convergence and complete convergence for weighted sums of AANA random variables

Let {X_n, n ? 1} be a sequence of asymptotically almost negatively associated (AANA, for short) random variables which is stochastically dominated by a random variable X, and {d_ni, 1 ? i ? n, n ? 1} be a sequence of real function, which is defined on a compact set E. Under some suitable conditions, we investigate some convergence properties for weighted sums of AANA random variables, especially the L^p convergence and the complete convergence. As an application, the Marcinkiewicz–Zygmund-type strong law of large numbers for AANA random variables is obtained.     

The probability distribution and the expected value of a stopping variable associated with one-sided cusum procedures for non-negative integer valued random variables

The structure of a stopping variable N based on one-sided CUSUM procedures is analyzed. Stopping occurs when a Markovian sequence of maxima of partial sums {M } crosses a certain boundary. On the basis of a recursive relationship between the M_n+1 and M_n a recursive equation is derived for the determination of the defective distributions K_n(x) = P{M ≤ x, N ≤n} . This recursive equation yields a recursive algorithm for the determination of P {N > n} . The paper studies the case when the basic random variables are non-negative integers-valued. In these cases the values of P{N > n} and E{N} can be determined by solving proper systems of linear equations.     

Moments of suprema of random variables

The supremum of random variables representing a sequence of rewards is of interest in establishing the existence of optimal stopping rules. Necessary and sufficient conditions are given for existence of moments of sup_n(X_n ? c_n) and sup_n(S_n ? c_n) where X₁, X₂, … are i.i.d. random variables, S_n = X₁ + … + X_n, and c_n = (nL(n))^1/r, 0 < r < 2, L = 1, L = log, and L = log log. Following Cohn (1974), “rates of convergence” results are used in the proof.     

A zero-one law for large order statistics

Fix r ≥ 1, and let {M_nr} be the rth largest of {X₁,X₂,…X_n}, where X₁,X₂,… is a sequence of i.i.d. random variables with distribution function F. It is proved that P[M_nr ≤ u_n i.o.] = 0 or 1 according as the series Σ^∞_n=3Fⁿ(u_n)(log log n)^r/n converges or diverges, for any real sequence {u_n} such that n{1 -F(u_n)} is nondecreasing and divergent. This generalizes a result of Bamdorff-Nielsen (1961) in the case r = 1.     

Record values and the geometric distribution

Suppose {X_n, n≥1} is a sequence of independent and identically distributed discrete random variables having the common distribution function F(x). The exact distribution of the n-th record value is given under the assumption that F(x) has the geometric distribution. Various properties of the record values and some new characterizations of the geometric distribution are presented.     

Precise Large Deviations for Sums of Independent Random Variables with Consistently Varying Tails

In this article, we investigate the precise large deviations for a sum of independent but not identical distributed random variables. {X _n , n ≥ 1} are independent non-negative random variables with distribution functions {F _n , n ≥ 1}. We assume that the average of right tails of distribution functions F _n is equivalent to some distribution function F with consistently varying tails. In applications, we apply our main results to a realistic example (Pareto-type distribution) and obtain a specific result.     

Sequential Estimation of Expectations in the Presence of Trend2

A sequence of independent random variables {Z_n:n≥ 1} with unknown probability distributions is considered and the problem of estimating their expectations {M_n+1: n≥ 1} is examined. The estimation of M_n+1 is based on a finite set {z_k:1≤k≤n}, each z_k being an observed value of Z_k, 1 ≤k≤n, and also based on the assumption that {M_n:n≥ 1} follows an unknown trend of a specified form.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

Probability inequalities for continuous unimodal random variables with finite support

A paramecer-free Bernstein-type upper bound is derived for the probability that the sum S of n i.i.d, unimodal random variables with finite support, X₁ ,X₂,…,X_n, exceeds its mean E(S) by the positive value nt. The bound for P{S - nμ ≥ nt} depends on the range of the summands, the sample size n, the positive number t, and the type of unimodality assumed for X_i. A two-sided Gauss-type probability inequality for sums of strongly unimodal random variables is also given. The new bounds are contrasted to Hoeffding's inequality for bounded random variables and to the Bienayme-Chebyshev inequality. Finally, the new inequalities are applied to a classic probability inequality example first published by Savage (1961).     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

Lois limites pour les statistiques d'ordre dans le cas non identiquement distribue

Let X₁, X₂,… be a sequence of independent random variables with distribution functions F₁, where 1 ≤ i ≤ n, and for each n ≥ 1 let X_1,n ≤… ≤ X_n,n denote the order statistics of the first n random variables. Under suitable hypotheses about the F₁, we characterize the limit distribution functions H(x) for which P(X_k,n ? a_nx + b_n) → H(x), where a_n > 0 and b_n are real constants. We consider the cases where κ = κ(n) satisfies √n {κ(n)/n — λ} → 0 and √n {κ(n)/n — λ} → ∞ separately.