Modern Stochastics: Theory and Applications

Quasi-mixing limits of the killed symmetric Lévy process are studied. It is proved that (intrinsic) ultracontractivity of the underlying process implies the existence of its (uniformly) exponentially quasi-mixing limits. As a by-product, this implication ensures that the process has (uniformly) exponential quasi-ergodicity and (uniformly) exponentially fractional quasi-ergodicity on ${L^{p}}$ ($p\ge 1$). It is noteworthy that precise rates of convergence and precise limiting equalities are provided, which are determined by spectral gaps and eigenfunction ratios of the underlying process. Finally, three examples are provided to demonstrate the theoretical results.

Finite mixtures with different regression models for different mixture components naturally arise in statistical analysis of biological and sociological data. In this paper a model of mixtures with varying concentrations is considered in which the mixing probabilities are different for different observations. The modified local linear regression estimator (mLLRE) is considered for nonparametric estimation of the unknown regression function for the given component of mixture. The asymptotic normality of the mLLRE is proved in the case when the regressor’s probability density function has jumps. Theoretically optimal bandwidth is derived. Simulations were made to estimate the accuracy of the normal approximation.

Let ${({\xi _{k}},{\eta _{k}})_{k\ge 1}}$ be independent identically distributed random vectors with arbitrarily dependent positive components and ${T_{k}}:={\xi _{1}}+\cdots +{\xi _{k-1}}+{\eta _{k}}$ for $k\in \mathbb{N}$. The random sequence ${({T_{k}})_{k\ge 1}}$ is called a (globally) perturbed random walk. Consider a general branching process generated by ${({T_{k}})_{k\ge 1}}$ and let ${Y_{j}}(t)$ denote the number of the jth generation individuals with birth times $\le t$. Assuming that $\mathrm{Var}\hspace{0.1667em}{\xi _{1}}\in (0,\infty )$ and allowing the distribution of ${\eta _{1}}$ to be arbitrary, a law of the iterated logarithm (LIL) is proved for ${Y_{j}}(t)$. In particular, an LIL for the counting process of ${({T_{k}})_{k\ge 1}}$ is obtained. The latter result was previously established in the article by Iksanov, Jedidi and Bouzeffour (2017) under the additional assumption that $\mathbb{E}{\eta _{1}^{a}}\lt \infty $ for some $a\gt 0$. In this paper, it is shown that the aforementioned additional assumption is not needed.