Related papers: Strong approximations for long memory sequences ba…
We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element…
In this paper we study the asymptotic behaviour of weighted random sums when the sum process converges stably in law to a Brownian motion and the weight process has continuous trajectories, more regular than that of a Brownian motion. We…
The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer…
Let $X, X_1, X_2,...$ be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in $D[0, 1]$ for the partial sum processes $\{S_{[nt]}, 0\le t\le 1\}$, where…
We consider Riemann sum approximations of stochastic integrals with respect to the fractional Browian motion of index $H\geq \frac12$. We show the convergence of these schemes at first and second order. The processes obtained in the limit…
We present a random walk approximation to fractional Brownian motion where the increments of the fractional random walk are defined as a weighted sum of the past increments of a Bernoulli random walk.
Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric alpha-stable processes are used to construct new examples of processes which exhibit both divergent…
Approximations of fractional Brownian motion using Poisson processes whose parameter sets have the same dimensions as the approximated processes have been studied in the literature. In this paper, a special approximation to the…
We investigate approximation of a Bernoulli partial sum process to the accompanying Poisson process in the non-i.i.d. case. The rate of closeness is studied in terms of the minimal distance in probability.
In this paper we estimate the rest of the approximation of a stationary process by a martingale in terms of the projections of partial sums. Then, based on this estimate, we obtain almost sure approximation of partial sums by a martingale…
Hermite processes are self--similar processes with stationary increments which appear as limits of normalized sums of random variables with long range dependence. The Hermite process of order $1$ is fractional Brownian motion and the…
In this work we introduce correlated random walks on $\Z$. When picking suitably at random the coefficient of correlation, and taking the average over a large number of walks, we obtain a discrete Gaussian process, whose scaling limit is…
In this paper we study the convergence to fractional Brownian motion for long memory time series having independent innovations with infinite second moment. For the sake of applications we derive the self-normalized version of this theorem.…
This paper reviews and extends some recent results on the multivariate fractional Brownian motion (mfBm) and its increment process. A characterization of the mfBm through its covariance function is obtained. Similarly, the correlation and…
We study well-posedness of sweeping processes with stochastic perturbations generated by a fractional Brownian motion and convergence of associated numerical schemes. To this end, we first prove new existence, uniqueness and approximation…
We study the asymptotic behavior of wavelet coefficients of random processes with long memory. These processes may be stationary or not and are obtained as the output of non--linear filter with Gaussian input. The wavelet coefficients that…
Probabilistic graphical models are a key tool in machine learning applications. Computing the partition function, i.e., normalizing constant, is a fundamental task of statistical inference but it is generally computationally intractable,…
In this paper we study approximations for boundary crossing probabilities for the moving sums of i.i.d. normal random variables. We propose approximating a discrete time problem with a continuous time problem allowing us to apply developed…
In this paper we study strong approximations (invariance principles) of the sequential uniform and general Bahadur--Kiefer processes of long-range dependent sequences. We also investigate the strong and weak asymptotic behavior of the…
The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for…