Bojan Basrak

Regular variation of GARCH processes

We show that the finite-dimensional distributions of a GARCH process are regularly varying, i.e., the tails of these distributions are Pareto-like and hence heavy-tailed. Regular variation of the joint distributions provides insight into the moment properties of the process as well as the dependence structure between neighboring observations when both are large. Regular variation also plays a vital role in establishing the large sample behavior of a variety of statistics from a GARCH process including the sample mean and the sample autocovariance and autocorrelation functions. In particular, if the 4th moment of the process does not exist, the rate of convergence of the sample autocorrelations becomes extremely slow, and if the second moment does not exist, the sample autocorrelations have non-degenerate limit distributions.

Plasticity of the Streptomyces Genome-Evolution and Engineering of New Antibiotics

Streptomyces is a genus of soil dwelling bacteria with the ability to produce natural products that have found widespread use in medicine. Annotation of Streptomyces genome sequences has revealed far more biosynthetic gene clusters than previously imagined, offering exciting possibilities for future combinatorial biosynthesis. Experiments to manipulate modular biosynthetic clusters to create novel chemistries often result in no detectable product or product yield is extremely low. Understanding the coupling between components in these hybrid enzymes will be crucial for efficient synthesis of new compounds. We are using new algebraic approaches to predict protein properties, and homologous recombination to exploit natural evolutionary constraints to generate novel functional enzymes. The methods and techniques developed could easily be adapted to study modular, multi-interacting complex systems where appreciable biochemical and comparative sequence data are available, for example, clinically significant non-ribosomally synthesised peptides and polyketides.

The sample ACF of a simple bilinear process

We consider a simple bilinear process Xt=aXt-1+bXt-1Zt-1+Zt, where (Zt) is a sequence of iid N(0,1) random variables. It follows from a result by Kesten (1973, Acta Math. 131, 207-248) that Xt has a distribution with regularly varying tails of index [alpha]>0 provided the equation Ea+bZ1u=1 has the solution u=[alpha]. We study the limit behaviour of the sample autocorrelations and autocovariances of this heavy-tailed non-linear process. Of particular interest is the case when [alpha]

A complete convergence theorem for stationary regularly varying multivariate time series

For a class of stationary regularly varying and weakly dependent time series, we prove the so-called complete convergence result for the corresponding space-time point processes. As an application of our main theorem, we give a simple proof of the invariance principle for the corresponding partial maximum process.

Clustering of protein domains for functional and evolutionary studies

BackgroundThe number of protein family members defined by DNA sequencing is usually much larger than those characterised experimentally. This paper describes a method to divide protein families into subtypes purely on sequence criteria. Comparison with experimental data allows an independent test of the quality of the clustering.ResultsAn evolutionary split statistic is calculated for each column in a protein multiple sequence alignment; the statistic has a larger value when a column is better described by an evolutionary model that assumes clustering around two or more amino acids rather than a single amino acid. The user selects columns (typically the top ranked columns) to construct a motif. The motif is used to divide the family into subtypes using a stochastic optimization procedure related to the deterministic annealing EM algorithm (DAEM), which yields a specificity score showing how well each family member is assigned to a subtype. The clustering obtained is not strongly dependent on the number of amino acids chosen for the motif. The robustness of this method was demonstrated using six well characterized protein families: nucleotidyl cyclase, protein kinase, dehydrogenase, two polyketide synthase domains and small heat shock proteins. Phylogenetic trees did not allow accurate clustering for three of the six families.ConclusionThe method clustered the families into functional subtypes with an accuracy of 90 to 100%. False assignments usually had a low specificity score.

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.

On randomly spaced observations and continuous-time random walks

We consider random variables observed at arrival times of a renewal process, which possibly depends on those observations and has regularly varying steps with infinite mean. Due to the dependence and heavy tailed steps, the limiting behavior of extreme observations until a given time

Efficient estimation in the semiparametric normal regression-copula model with a focus on QTL mapping

t

Regularly varying multivariate time series

tends to be rather involved. We describe this asymptotics and generalize several partial results which appeared in this setting. In contrast to the earlier studies, our analysis is based in the point processes theory. The theory is applied to determine the asymptotic distribution of maximal excursions and sojourn times for continuous time random walks.

The sample autocorrelation function of non-linear time series

The semiparametric normal copula model is studied with a correlation matrix that depends on a covariate. The bivariate version of this regression-copula model has been proposed for statistical analysis of Quantitative Trait Loci (QTL) via twin data. Appropriate linear combinations of Van der Waerden’s normal scores rank correlation coefficients yield