Modern Stochastics: Theory and Applications

We investigate the fractional Riemann-Liouville integral of a general continuous Gaussian process, focusing first on the functional weak convergence of suitably rescaled processes. Building on these results and recent theoretical advances, we deduce functional large deviation principles. For small times, the asymptotic behavior depends solely on the covariance of the underlying process at zero, while for large times, appropriate rescaling of the covariance is required and additional regularity assumptions must be imposed. A central aspect of our work is the explicit characterization of the reproducing kernel Hilbert spaces of the fractionally integrated processes, including concrete formulas for the related norms. As a notable special case, when the underlying Gaussian process is Gauss-Markov, the reproducing kernel Hilbert spaces and the covariance structure can be described explicitly, providing precise insights into the corresponding rate functions.

We study a level-dependent PH/PH/1 queue with an infinite waiting room and derive explicit density formulas for the associated Matrix Mittag-Leffler (MML) random variables. The motivation arises from the use of Phase Type (PH) and Mittag-Leffler type distributions in modeling heavy-tailed and non-Markovian arrival and service processes in queueing systems, where classical exponential assumptions may fail to capture realistic behavior. Our goal is to obtain explicit density expressions for the MML random variables associated with the truncated queueing systems PH/M/$1\sim n$ and M/PH/$1\sim n$, where n denotes the truncation level at which the queueing process is absorbed. We first analyze PH/M/1 queues in which the inter-arrival times are PH-distributed, focusing specifically on exponential and Erlang cases, and derive closed-form MML densities. We then consider M/PH/1 queues with PH-distributed service times and obtain analogous explicit density formulas. Several examples with PH-generators of different orders are presented to illustrate the results.

We study a time-changed variant of the Erlang queue by taking the first hitting time of a mixed stable subordinator as the time-changing component. We call it the mixed time-changed Erlang queue. We derive the system of fractional differential equations that governs its state probabilities. The explicit expressions for the state probabilities of mixed time-changed Erlang queue and their Laplace transform are derived. Also, an equivalent representation of this time-changed queue in terms of phases is provided, and its mean queue length is obtained. Some of its distributional properties such as the distribution of its inter-arrival times, inter-phase times, service times and busy period are derived. Later, its conditional waiting time is discussed and some plots of sample paths simulation are presented.

Distribution of the sum of independent identically distributed symmetric lattice vectors is approximated by the accompanying compound Poisson law and the second-order Hipp-type signed compound Poisson measure. Bergström-type asymptotic expansion is constructed. The accuracy of approximation is estimated in the total variation metric and, in many cases, is of the order $O({n^{-1}})$.