Rafal Kulik

Long-memory processes : Probabilistic Properties and Statistical Methods

Definition of Long Memory.- Origins and Generation of Long Memory.- Mathematical Concepts.- Limit Theorems.- Statistical Inference for Stationary Processes.- Statistical Inference for Nonlinear Processes.- Statistical Inference for Nonstationary Processes.- Forecasting.- Spatial and Space-Time Processes.- Resampling.- Function Spaces.- Regularly Varying Functions.- Vague Convergence.- Some Useful Integrals.- Notation and Abbreviations.- References. ai

Wavelet regression in random design with heteroscedastic dependent errors

We investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise. Adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of Besov scales, f ∈ B S π,r , and for a variety of L P error measures. We consider error distributions with Long-Range-Dependence parameter α, 0 2, it is seen that there are three rate phases, namely the dense, sparse and long range dependence phase, depending on the relative values of s, p, π and α. Furthermore, we show that long range dependence does not come into play for shape estimation f - ∫ f. The theory is illustrated with some numerical examples.

Spatiotemporal distribution of Holocene populations in North America

Significance We provide the first maps to our knowledge of spatiotemporal paleodemographic growth following human migration into the Americas for the past 13,000 y, using a statistical approach that simultaneously addresses sampling and taphonomic biases. The Canadian Archaeological Radiocarbon Database is sufficiently complete in many areas, demonstrating high correspondence between continental-scale 14C-inferred population estimates and generally accepted archaeological history. Increases in population density seem robust for eastern and western North America, as well as central Alaska and the region surrounding Cahokia. These results are the first step toward being able to understand continental-scale human impacts on the North American ecosystem during the Holocene as well as demographic growth and migrations in relation to environmental changes. As the Cordilleran and Laurentide Ice Sheets retreated, North America was colonized by human populations; however, the spatial patterns of subsequent population growth are unclear. Temporal frequency distributions of aggregated radiocarbon (14C) dates are used as a proxy of population size and can be used to track this expansion. The Canadian Archaeological Radiocarbon Database contains more than 35,000 14C dates and is used in this study to map the spatiotemporal demographic changes of Holocene populations in North America at a continental scale for the past 13,000 y. We use the kernel method, which converts the spatial distribution of 14C dates into estimates of population density at 500-y intervals. The resulting maps reveal temporally distinct, dynamic patterns associated with paleodemographic trends that correspond well to genetic, archaeological, and ethnohistoric evidence of human occupation. These results have implications for hypothesizing and testing migration routes into and across North America as well as the relative influence of North American populations on the evolution of the North American ecosystem.

Multichannel deconvolution with long range dependence: Upper bounds on the Lp-risk (1≤p<∞)

Abstract We consider multichannel deconvolution in a periodic setting with long-memory errors under three different scenarios for the convolution operators, i.e., super-smooth, regular-smooth and box-car convolutions. We investigate global performances of linear and hard-thresholded non-linear wavelet estimators for functions over a wide range of Besov spaces and for a variety of loss functions defining the risk. In particular, we obtain upper bounds on convergence rates using the L p -risk ( 1 ≤ p ∞ ) . Contrary to the case where the errors follow independent Brownian motions, it is demonstrated that multichannel deconvolution with errors that follow independent fractional Brownian motions with different Hurst parameters results in a much more involved situation. An extensive finite-sample numerical study is performed to supplement the theoretical findings.

Limit theorems for long memory stochastic volatility models with infinite variance: Partial Sums and Sample Covariances.

In this paper we extend the existing literature on the asymptotic behavior of the partial sums and the sample covariances of long-memory stochastic volatility models in the case of infinite variance. We also consider models with leverage, for which our results are entirely new in the infinite-variance case. Depending on the interplay between the tail behavior and the intensity of dependence, two types of convergence rates and limiting distributions can arise. In particular, we show that the asymptotic behavior of partial sums is the same for both long memory in stochastic volatility and models with leverage, whereas there is a crucial difference when sample covariances are considered.

Strong approximations for long memory sequences based partial sums, counting and their Vervaat processes

We study the asymptotic behaviour of partial sums of long range dependent random variables and that of their counting process, together with an appropriately normalized integral process of the sum of these two processes, the so-called Vervaat process. The first two of these processes are approximated by an appropriately constructed fractional Brownian motion, while the Vervaat process in turn is approximated by the square of the same fractional Brownian motion.

Heavy-Tailed Branching Process with Immigration

In this article, we analyze a branching process with immigration defined recursively by X t = θ t ○ X t−1 + B t for a sequence (B t ) of i.i.d. random variables and random mappings , with being a sequence of ℕ0-valued i.i.d. random variables independent of B t . We assume that one of generic variables A and B has a regularly varying tail distribution. We identify the tail behavior of the distribution of the stationary solution X t . We also prove CLT for the partial sums that could be further generalized to FCLT. Finally, we also show that partial maxima have a Fréchet limiting distribution.

Tail behaviour of the area under a random process, with applications to queueing systems, insurance and percolations

The areas under the workload process and under the queueing process in a single-server queue over the busy period have many applications not only in queueing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.

Statistical Inference for Nonstationary Processes

In this chapter, statistical inference for nonstationary processes is discussed. For long-memory, or, more generally, fractional stochastic processes this is of particular interest because long-range dependence often generates sample paths that mimic certain features of nonstationarity. It is therefore often not easy to distinguish between stationary long-memory behaviour and nonstationary structures. For statistical inference, including estimation, testing and forecasting, the distinction between stationary and nonstationary, as well as between stochastic and deterministic components, is essential.

Statistical Inference for Stationary Processes

This chapter deals with statistical inference for long-range dependent linear and subordinated processes. Some of the tools will also be used in Chaps. 6 and 7 when we shall consider corresponding problems for nonlinear and nonstationary long-memory time series.