One Dimensional Discrete Scan Statistics for Dependent Models and Some Related Problems

The one dimensional discrete scan statistic is considered over sequences of random variables generated by block factor dependence models. Viewed as a maximum of an 1-dependent stationary sequence, the scan statistics distribution is approximated with accuracy and sharp bounds are provided. The longest increasing run statistics is related to the scan statistics and its distribution is studied. The moving average process is a particular case of block factor and the distribution of the associated scan statistics is approximated. Numerical results are presented.

SUPERPOSITIONED STATIONARY COUNT TIME SERIES

This paper probabilistically explores a class of stationary count time series models built by superpositioning (or otherwise combining) independent copies of a binary stationary sequence of zeroes and ones. Superpositioning methods have proven useful in devising stationary count time series having prespecified marginal distributions. Here, basic properties of this model class are established and the idea is further developed. Specifically, stationary series with binomial, Poisson, negative binomial, discrete uniform, and multinomial marginal distributions are constructed; other marginal distributions are possible. Our primary goal is to derive the autocovariance function of the resulting series.

On exponential decay of the variance of BLUE for the mean of a stationary sequence

In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of the variance of the best linear unbiased estimator (BLUE) for the unknown mean of a stationary sequence possessing a spectral density. In particular, we show that a necessary condition for variance of BLUE to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero.

Functional Central Limit Theorem for Empirical Processes

Here we discuss the Gaussian approximation for the empirical process under different kinds of dependence assumptions for the underlying stationary sequence. First, we state a general criterion to prove tightness of the empirical process associated with a stationary sequence of uniformly distributed random variables. This tightness criterion can be verified for many different dependence structures. For ρ‎-mixing sequences, by an application of a Rosenthal-type inequality, tightness is verified under the same condition leading to the usual CLT. For α‎-dependent sequences whose α‎-dependent coefficients decay polynomially to zero, it is shown to hold with the help of the Rosenthal inequality stated in Section 3.3. Since the asymptotic behavior of the finite-dimensional distributions of the empirical process is handled via the CLT developed in previous chapters, we then derive the functional CLT for the empirical process associated with the above-mentioned classes of stationary sequences. β‎-dependent sequences are also investigated by directly proving tightness of the empirical process.

Random Walk in Random Scenery

In this chapter we investigate the question of central limit behavior and its functional form for the partial sums associated with a centered L2-stationary sequence of real-valued random variables (usually called the random scenery) sampled by a recurrent one-dimensional strongly aperiodic random walk. This question is handled under various conditions dependent on the random scenery. In particular, we assume that the random scenery either satisfies an asymptotic negative dependence condition, or is a function of a determinantal process and a Gaussian sequence, or satisfies a mild projective criterion. We first show that study of central limit behavior for such random walks in random scenery can be handled with results related to linear statistics developed in Chapter 12, provided the random walk has good properties. We then look extensively at the properties of a recurrent one-dimensional strongly aperiodic random walk. The functional form of the central limit theorem is also investigated.

Examples and Counterexamples

In this chapter we further comment on the sharpness of several results presented in this monograph, by presenting examples and counterexamples. We study first the moment properties of the renewal Markov chain introduced in Chapter 11. This allows us to show that the Maxwell–Woodroofe projective condition introduced in Chapter 4 is essentially optimal for the partial sums of a stationary sequence in L2 to satisfy the central limit theorem under the standard normalization √n. Moreover, we also investigate the sharpness of the Burkholder-type inequality developed in Chapter 3 via Maxwell–Woodroofe-type characteristics. In the last part of this chapter, we analyze several telescopic-type examples allowing us to elucidate the fact that a CLT behavior does not imply its functional form under any normalization. Even in the case when the variance of the partial sums is linear in n, the CLT does not necessarily imply the invariance principle.

Almost sure convergence for self-normalized products of sums of partial sums of ρ¯-mixing sequences

Let X,X1,X2,... be a stationary sequence of ??-mixing positive random variables. A universal result in the area of almost sure central limit theorems for the self-normalized products of sums of partial sums (?kj =1(Tj/(j(j+1)?/2)))?=(?Vk) is established, where: Tj = ?ji=1 Si,Si = ?i k=1 Xk,Vk = ??ki=1 X2i,? = EX, ? > 0. Our results generalize and improve those on almost sure central limit theorems obtained by previous authors from the independent case to ??-mixing sequences and from partial sums case to self-normalized products of sums of partial sums.