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.

Probabilistic properties of vantage point trees are studied. A vp-tree built from a sequence of independent identically distributed points in ${[-1,\hspace{0.1667em}1]^{d}}$ with the ${\ell _{\infty }}$-distance function is considered. The length of the leftmost path in the tree, as well as partitions over the space it produces are analyzed. The results include several convergence theorems regarding these characteristics, as the number of nodes in the tree tends to infinity.

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.

The insurance model when the amount of claims depends on the state of the insured person (healthy, ill, or dead) and claims are connected in a Markov chain is investigated. The signed compound Poisson approximation is applied to the aggregate claims distribution after $n\in \mathbb{N}$ periods. The accuracy of order $O({n^{-1}})$ and $O({n^{-1/2}})$ is obtained for the local and uniform norms, respectively. In a particular case, the accuracy of estimates in total variation and non-uniform estimates are shown to be at least of order $O({n^{-1}})$. The characteristic function method is used. The results can be applied to estimate the probable loss of an insurer to optimize an insurance premium.

The paper deals with bonus–malus systems with different claim types and varying deductibles. The premium relativities are softened for the policyholders who are in the malus zone and these policyholders are subject to per claim deductibles depending on their levels in the bonus–malus scale and the types of the reported claims. We introduce such bonus–malus systems and study their basic properties. In particular, we investigate when it is possible to introduce varying deductibles, what restrictions we have and how we can do this. Moreover, we deal with the special case where varying deductibles are applied to the claims reported by policyholders occupying the highest level in the bonus–malus scale and consider two allocation principles for the deductibles. Finally, numerical illustrations are presented.