Modern Stochastics: Theory and Applications

The time-inhomogeneous autoregressive model AR(1) is studied, which is the process of the form ${X_{n+1}}={\alpha _{n}}{X_{n}}+{\varepsilon _{n}}$, where ${\alpha _{n}}$ are constants, and ${\varepsilon _{n}}$ are independent random variables. Conditions on ${\alpha _{n}}$ and distributions of ${\varepsilon _{n}}$ are established that guarantee the geometric recurrence of the process. This result is applied to estimate the stability of n-steps transition probabilities for two autoregressive processes ${X^{(1)}}$ and ${X^{(2)}}$ assuming that both ${\alpha _{n}^{(i)}}$, $i\in \{1,2\}$, and distributions of ${\varepsilon _{n}^{(i)}}$, $i\in \{1,2\}$, are close enough.

The factorial moments of any Markov branching process describe the behaviour of its probability generating function $F(t,s)$ in the neighbourhood of the point $s=1$. They are applied to solve the forward Kolmogorov equation for the critical Markov branching process with geometric reproduction of particles. The solution includes quickly convergent recurrent iterations of polynomials. The obtained results on factorial moments enable computation of statistical measures as shape and skewness. They are also applicable to the comparison between critical geometric branching and linear birth-death processes.

Recurrence times and the number of renewals in $(0,t]$ are fundamental quantities in renewal theory. Firstly, it is proved that the upper orthant order for the pair of the forward and backward recurrence times may result in NWUC (NBUC) interarrivals. It is also demonstrated that, under DFR interarrival times, the backward recurrence time is smaller than the forward recurrence time in the hazard rate order. Lastly, the sign of the covariance between the forward recurrence time and the number of renewals in $(0,t]$ at a fixed time point t and when $t\to \infty $ is studied assuming that the interarrival distribution belongs to certain ageing classes.

Generalizing earlier work of Delbaen and Haezendonck for given compound renewal process S under a probability measure P we characterize all probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound renewal process under Q. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.

The main subject of the study in this paper is the simultaneous renewal time for two time-inhomogeneous Markov chains which start with arbitrary initial distributions. By a simultaneous renewal we mean the first time of joint hitting the specific set C by both processes. Under the condition of existence a dominating sequence for both renewal sequences generated by the chains and non-lattice condition for renewal probabilities an upper bound for the expectation of the simultaneous renewal time is obtained.

Let $(X_{k},\xi _{k})_{k\in \mathbb{N}}$ be a sequence of independent copies of a pair $(X,\xi )$ where X is a random process with paths in the Skorokhod space $D[0,\infty )$ and ξ is a positive random variable. The random process with immigration $(Y(u))_{u\in \mathbb{R}}$ is defined as the a.s. finite sum $Y(u)=\sum _{k\ge 0}X_{k+1}(u-\xi _{1}-\cdots -\xi _{k})\mathbb{1}_{\{\xi _{1}+\cdots +\xi _{k}\le u\}}$. We obtain a functional limit theorem for the process $(Y(ut))_{u\ge 0}$, as $t\to \infty $, when the law of ξ belongs to the domain of attraction of an α-stable law with $\alpha \in (0,1)$, and the process X oscillates moderately around its mean $\mathbb{E}[X(t)]$. In this situation the process $(Y(ut))_{u\ge 0}$, when scaled appropriately, converges weakly in the Skorokhod space $D(0,\infty )$ to a fractionally integrated inverse stable subordinator.

In this paper, we consider two time-inhomogeneous Markov chains ${X_{t}^{(l)}}$, $l\in \{1,2\}$, with discrete time on a general state space. We assume the existence of some renewal set C and investigate the time of simultaneous renewal, that is, the first positive time when the chains hit the set C simultaneously. The initial distributions for both chains may be arbitrary. Under the condition of stochastic domination and nonlattice condition for both renewal processes, we derive an upper bound for the expectation of the simultaneous renewal time. Such a bound was calculated for two time-inhomogeneous birth–death Markov chains.