Related papers: Multivariate limits of multilinear polynomial-form…

We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one…

Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the…

Over the last 30 years, extensive work has been devoted to developing central limit theory for partial sums of subordinated long memory linear time series. A much less studied problem, motivated by questions that are ubiquitous in extreme…

We derive theorems which outline explicit mechanisms by which anomalous scaling for the probability density function of the sum of many correlated random variables asymptotically prevails. The results characterize general anomalous scaling…

Let $R$ be a commutative ring, $f \in R[X_1,\ldots,X_k]$ a multivariate polynomial, and $G$ a finite subgroup of the group of units of $R$ satisfying a certain constraint, which always holds if $R$ is a field. Then, we evaluate $\sum…

This paper is devoted to parameter estimation for partially observed polynomial state space models. This class includes discretely observed affine or more generally polynomial Markov processes. The polynomial structure allows for the…

For a linearly recurrent vector sequence P[n+1] = A(n) * P[n], consider the problem of calculating either the n-th term P[n] or L<=n arbitrary terms P[n_1],...,P[n_L], both for the case of constant coefficients A(n)=A and for a matrix A(n)…

We introduce a multidimensional walk with memory and random tendency. The asymptotic behaviour is characterized, proving a law of large numbers and showing a phase transition from diffusive to superdiffusive regimes. In first case, we…

We find that multifractal scaling is a robust property of a large class of continuous stochastic processes, constructed as exponentials of long-memory processes. The long memory is characterized by a power law kernel with tail exponent…

Continuing our recent work we study polynomial masks of multivariate tight wavelet frames from two additional and complementary points of view: convexity and system theory. We consider such polynomial masks that are derived by means of the…

We define multifractional Hermite processes which generalize and extend both multifractional Brownian motion and Hermite processes. It is done by substituting the Hurst parameter in the definition of Hermite processes as a multiple…

Considering commutator monomials of the non-commutative associative variables $X_1,\ldots,X_n$; we determine the maximal possible number of alternating associative monomials in their noncommutative polynomial expansions. This is achieved by…

Linear processes are defined as a discrete-time convolution between a kernel and an infinite sequence of i.i.d. random variables. We modify this convolution by introducing decimation, that is, by stretching time accordingly. We then…

We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class…

It is generally accepted that many time series of practical interest exhibit strong dependence, i.e., long memory. For such series, the sample autocorrelations decay slowly and log-log periodogram plots indicate a straight-line…

We study subsampling estimators for the limit variance \[ \sigma^2=Var(X_1)+2 \sum_{k=2}^\infty Cov(X_1,X_k) \] of partial sums of a stationary stochastic process $(X_k)_{k\geq 1}$. We establish $L_2$-consistency of a non-overlapping block…

We consider sequences of random variables of the type $S_n= n^{-1/2} \sum_{k=1}^n \{f(X_k)-\E[f(X_k)]\}$, $n\geq 1$, where $X=(X_k)_{k\in \Z}$ is a $d$-dimensional Gaussian process and $f: \R^d \rightarrow \R$ is a measurable function. It…

An $(n, k)$-Poisson Multinomial Distribution (PMD) is a random variable of the form $X = \sum_{i=1}^n X_i$, where the $X_i$'s are independent random vectors supported on the set of standard basis vectors in $\mathbb{R}^k.$ In this paper, we…

Let $d, r \in \N$, $\|\cdot\|$ any norm on $\R^d$ and $B$ denote the unit ball with respect to this norm. We show that any sequence $v_1,v_2,...$ of vectors in $B$ can be partitioned into $r$ subsequences $V_1, ..., V_r$ in a balanced…

We prove a central limit theorem for random sums of the form $\sum_{i=1}^{N_n} X_i$, where $\{X_i\}_{i \geq 1}$ is a stationary $m-$dependent process and $N_n$ is a random index independent of $\{X_i\}_{i\geq 1}$. Our proof is a…