Alfredas Račkauskas

The Accuracy of Gaussian Approximation in Banach Spaces

Let B be a real separable Banach space with norm || · || = || · || B . Suppose that X, X 1, X 2, … ∈ B are independent and identically distributed (i.i.d.) random elements (r.e.’s) taking values in B. Furthermore, assume that EX = 0 and that there exists a zero-mean Gaussian r.e. Y ∈ B such that the covariances of X and Y coincide.

Invariance principles for adaptive self-normalized partial sums processes

Let [zeta]nse be the adaptive polygonal process of self-normalized partial sums Sk=[summation operator]1[less-than-or-equals, slant]i[less-than-or-equals, slant]kXi of i.i.d. random variables defined by linear interpolation between the points (Vk2/Vn2,Sk/Vn), k[less-than-or-equals, slant]n, where Vk2=[summation operator]i[less-than-or-equals, slant]k Xi2. We investigate the weak Holder convergence of [zeta]nse to the Brownian motion W. We prove particularly that when X1 is symmetric, [zeta]nse converges to W in each Holder space supporting W if and only if X1 belongs to the domain of attraction of the normal distribution. This contrasts strongly with Lampertis FCLT where a moment of X1 of order p>2 is requested for some Holder weak convergence of the classical partial sums process. We also present some partial extension to the nonsymmetric case.

Functional data analysis for clients segmentation tasks

Functional analysis of variance (FANOVA) is used to test whether there are some effects in cash flow processes that could be attributed to some categorical variable. Two data sets are under investigation. First data set represents cash flow processes with geographical and the second one with merchant type attributes.

Large deviations for martingales with some applications

AbstractLet(Xi) be a martingale difference sequence. LetY be a standard normal random variable. We investigate the rate of uniform convergence

The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

P\left\{ {\sum\limits_{k = 1}^n {X_k } > \sqrt n r} \right\}/P\{ Y > r\} \to 0asn \to \infty ,

Necessary and sufficient condition for the Lamperti invariance principle

asn → ∞, over 0⩽r⩽o(n1/6) in the case of bounded martingale differences. The results are applied to prove large deviations for the ‘baker transformation’. Moderate deviations for martingales are also discussed.

OPERATOR FRACTIONAL BROWNIAN MOTION AS LIMIT OF POLYGONAL LINES PROCESSES IN HILBERT SPACE

Abstract For rather general moduli of smoothness ρ, like ρ(ℎ)=ℎ α ln β (𝑐/ℎ), we consider the Hölder spaces H ρ (B) of functions [0,1] d → B is a separable Banach space. We establish an isomorphism between H ρ (B) and some sequence Banach space. With this analytical tool, we follow a very natural way to study, in terms of second differences, the existence of a version in H ρ (B) for a given random field.

The central limit theorem for a sequence of random processes with space-varying long memory*

In this paper, we deal with the asymptotic distribution of the maximum increment of a random walk with a regularly varying jump size distribution. This problem is motivated by a long-standing problem on change point detection for epidemic alternatives. It turns out that the limit distribution of the maximum increment of the random walk is one of the classical extreme value distributions, the Frechet distribution. We prove the results in the general framework of point processes and for jump sizes taking values in a separable Banach space.