We establish large deviation estimates for the optimal filter where the observation process is corrupted by a fractional Brownian motion. The observation process is transformed to an equivalent model which is driven by a standard Brownian motion. The large deviations in turn are established by...

We establish the asymptotic normality of the sample principal components of functional stochastic processes under nonrestrictive assumptions which admit nonlinear functional time series models. We show that the aforementioned asymptotic depends only on the asymptotic normality of the sample...

We give a new sufficient condition of the quasi-Gibbs property. This result is a refinement of one given in a previous paper (Osada (in press) [18]), and will be used in a forthcoming paper to prove the quasi-Gibbs property of Airy random point fields (RPFs) and other RPFs appearing under...

In this paper, we derive transform formulae for linear functionals of affine processes and their bridges whose state space is the set of positive semidefinite d×d matrices. Particularly, we investigate the relationship between such transforms and certain integral equations. Our findings extend...

We consider the branching processes in random environment. In this paper, we deal with the case of environments which are chosen stationary and ergodic from the finite set of geometrical offspring distributions. We denote by Zn the population at the n-th generation. We show that the large...

We prove both the validity and the sharpness of the law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges.

Let {Xk,k∈Z} be a zero mean d-dimensional stationary process, and let Sn,d=(Sn,1,Sn,2,…,Sn,d)T with Sn,h=1n∑k=1nXk,h, where Xk,h denotes the single components of {Xk,k∈Z}. Under a weak dependence condition, we show that the ellipsoid Xd2=max1≤k≤d(2k)−1/2|Sn,kTΓk−1Sn,k−k|...

The existence of strong and weak càdlàg versions of a solution to a linear equation in a Hilbert space H, driven by a Lévy process taking values in a Hilbert space U↩H is established. The so-called cylindrical càdlàg property is investigated as well. A special emphasis is put on infinite...

The purpose of this paper is to derive the stochastic expansion of self-normalized-residual functionals stemming from a class of diffusion type processes observed at high frequency, where total observing period may or may not tend to infinity. The result enables us to construct some explicit...

Motivated by problems in functional data analysis, in this paper we prove the weak convergence of normalized partial sums of dependent random functions exhibiting a Bernoulli shift structure.