This work is supported by Bulgarian NFSI, grant No. MM–704/97The regenerative excursion process Z(t), t = 0, 1, 2, . . . is constructed by two independent sequences X = {Xi , i ≥ 1} and Z = {Ti , (Zi (t), 0 ≤ t < Ti ), i ≥ 1}. For the embedded alternating renewal process, with interarrival times Xi – the time for the installation and Ti – the time for the work, are proved some limit theorems for the spent worktime and the residual worktime, when at least one of the means of Xi and Ti is infinite. ...

Consider an irreducible, aperiodic and positive recurrent discrete time Markov chain (Xn,n ≥ 0) with invariant distribution μ. We shall investigate the long time behaviour of some functionals of the chain, in particular the additive functional $S_n={\sum }_{i=1}^{n}f\left(X_i\right)$ S n = ∑ i = 1 n f ( X i ) for a possibly non square integrable functionf. To this end we shall link ergodic properties of the chain to mixing properties, extending known results in the continuous time case. We will then use existing results of convergence...

Let ${(X_t,Y_t)}_{t\in 𝕋}$ be a discrete or continuous-time Markov process with state space $𝕏\times {ℝ}^d$ where $𝕏$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. ${(X_t,Y_t)}_{t\in 𝕋}$ is assumed to be a Markov additive process. In particular, this implies that the first component ${\left(X_t\right)}_{t\in 𝕋}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process ${\left(Y_t\right)}_{t\in 𝕋}$ is shown to satisfy the...

We consider the stochastic recursion $Xₙ=AₙX_{n-1}+Bₙ$ for Markov dependent coefficients (Aₙ,Bₙ) ∈ ℝ⁺ × ℝ. We prove the central limit theorem, the local limit theorem and the renewal theorem for the partial sums Sₙ = X₁+ ⋯ + Xₙ.

Let ${\left(S_n\right)}_{n\ge 0}$ be a $\mathbb{Z}$-random walk and ${\left({\xi }_x\right)}_{x\in \mathbb{Z}}$ be a sequence of independent and identically distributed $ℝ$-valued random variables, independent of the random walk. Let $h$ be a measurable, symmetric function defined on ${ℝ}^2$ with values in $ℝ$. We study the weak convergence of the sequence ${𝒰}_n,n\in \mathbb{N}$, with values in $D[0,1]$ the set of right continuous real-valued functions with left limits, defined by$$\sum _{i,j=0}^{\left[nt\right]}h({\xi }_{S_i},{\xi }_{S_j}),t\in [0,1].$$Statistical applications are presented, in particular we prove a strong law of large numbers for $U$-statistics indexed by a one-dimensional...

Let (Sn)n≥0 be a $\mathbb{Z}$-random walk and ${\left({\xi }_x\right)}_{x\in \mathbb{Z}}$ be a sequence of independent and identically distributed $ℝ$-valued random variables, independent of the random walk. Let h be a measurable, symmetric function defined on ${ℝ}^2$ with values in $ℝ$. We study the weak convergence of the sequence ${𝒰}_n,n\in \mathbb{N}$, with values in D[0,1] the set of right continuous real-valued functions with left limits, defined by $$\sum _{i,j=0}^{\left[nt\right]}h({\xi }_{S_i},{\xi }_{S_j}),t\in [0,1].$$ Statistical applications are presented, in particular we prove a strong law of large numbers for U-statistics indexed by...

Based on an analytical approach to the definition of multiplicative free convolution on probability measures on the nonnegative line ℝ+ and on the unit circle $$𝕋$$ we prove analogs of limit theorems for nonidentically distributed random variables in classical Probability Theory.

We prove stable limit theorems and one-sided laws of the iterated logarithm for a class of positive, mixing, stationary, stochastic processes which contains those obtained from nonintegrable observables over certain piecewise expanding maps. This is done by extending Darling–Kac theory to a suitable family of infinite measure preserving transformations.