An approximate maximum likelihood estimation for non-Gaussian non-minimum phase moving average processes

An approximate maximum likelihood procedure is proposed for the estimation of parameters in possibly nonminimum phase (noninvertible) moving average processes driven by independent and identically distributed non-Gaussian noise. Under appropriate conditions, parameter estimates that are solutions of likelihood-like equations are consistent and are asymptotically normal. A simulation study for MA(2) processes illustrates the estimation procedure.     

Complete convergence for moving average processes associated to heavy-tailed distributions and applications

The complete convergence is obtained for the moving average processes associated to heavy-tailed distributions via integral test. As the applications, two versions of Chover's law of the iterated logarithm are deduced.     

Bootstrap in moving average models

We prove that the bootstrap principle works very well in moving average models, when the parameters satisfy the invertibility condition, by showing that the bootstrap approximation of the distribution of the parameter estimates is accurate to the ordero(n ^−1/2) a.s. Some simulation studies are also reported.     

Convergence rates of tail probabilities for sums under dependence assumptions

Let {X,X1,X2,……}be a zero mean strictly stationary Ф-mixing sequence. Set Sn=∑n k=1 and f（x^p）=∑∞n=1 n^r-2P（｜Sn｜≥x^p√ES2nlog n）,When ε〉（√2）1/p,for p〉1/2 and r〉1,the conditions for ∫∞ε f（x^p）dx 〈∞ to hold is established, by using coupled methods together withstrong approximation, which are different from the traditional symmetrization and Hoffman-JФrgensen inequality.     

On prediction of integrated moving average processes

We shall consider the asymptotic properties of predictors with estimated coefficients for IMA processes and how to determine the order of predictors to minimize the error of prediction. For this purpose, the effect of the initial values on predictors is also considered.     

Precise asymptotics of complete moment convergence on moving average

Let {ξi,-∞i∞} be a doubly infinite sequence of identically distributed-mixing random variables with zero means and finite variances,{ai,-∞i∞} be an absolutely summable sequence of real numbers and X k =∑i=-∞+∞ aiξi+k be a moving average process.Under some proper moment conditions,the precise asymptotics are established for     

Convergence rates for probabilities of moderate deviations for moving average processes

The present paper first shows that, without any dependent structure assumptions for a sequence of random variables, the refined results of the complete convergence for the sequence is equivalent to the corresponding complete moment convergence of the sequence. Then this paper investigates the convergence rates and refined convergence rates （or complete moment convergence） for probabilities of moderate deviations of moving average processes. The results in this paper extend and generalize some well-known results.     

Determination of the probability of ultimate ruin by maximum entropy applied to fractional moments

In this work we present two different numerical methods to determine the probability of ultimate ruin as a function of the initial surplus. Both methods use moments obtained from the Pollaczek–Kinchine identity for the Laplace transform of the probability of ultimate ruin. One method uses fractional moments combined with the maximum entropy method and the other is a probabilistic approach that uses integer moments directly to approximate the density.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

On some limit theorems for the maximum of sums of independent random processes

We investigate conditions for the weak convergence of the maximum of sums of independent random processes in the space L _p and present several applications to the asymptotic analysis of certain ω ²-type statistics. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1664–1674, December, 2008.     

The almost sure central limit theorems for the maxima of sums under some new weak dependence assumptions

We prove the almost sure central limit theorems for the maxima of partial sums of r.v.’s under a general condition of dependence due to Doukhan and Louhichi. We will separately consider the centered sequences and the sequences with positive expected values.     

Strong approximation of partial sums under dependence conditions with application to dynamical systems

In this paper, we obtain precise rates of convergence in the strong invariance principle for stationary sequences of real-valued random variables satisfying weak dependence conditions including strong mixing in the sense of Rosenblatt (1956) [30] as a special case. Applications to unbounded functions of intermittent maps are given.     

Almost sure convergence of weighted partial sums

Sufficient conditions of covariance type are presented for weighted averages of random variables with arbitrary dependence structure to converge to 0, both for logarithmic and general weighting. As an application, an a.s. local limit theorem of Csáki, Földes and Révész is revisited and slightly improved.     

On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables

Let {Xni, 1 ≤ n,i ＜∞} be an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 ＜ an ↑∞. The limiting behavior of maximum partial sums 1/an max 1≤k≤n| kΣi=1 Xni| is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].