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.

Let ${B^{H}}$ be a fractional Brownian motion with Hurst index $\frac{1}{2}\lt H\lt 1$. In this paper, we consider the time fractional functional differential equation of the form where $\frac{3}{2}-H\lt \gamma \lt 1$, ${^{C}}{D_{t}^{\gamma }}$ denotes the Caputo derivative, and ${x_{t}}\in {\mathcal{C}_{r}}=\mathcal{C}([-r,0],\mathbb{R})$ with ${x_{t}}(u)=x(t+u)$, $u\in [-r,0]$. We prove the global existence and uniqueness of the solution of the equation and study its viability. As an application, we also discuss the existence of positive solutions.

In many practical systems, the load changes at the moments when random events occur, which are often modeled as arrivals in a Poisson process independent of the current load state. This modeling approach is widely applicable in areas such as telecommunications, queueing theory, and reliability engineering. This motivates the development of models that combine family-wise scaling with non-Gaussian driving mechanisms, capturing discontinuities or jump-type behavior. In this paper, a stationary time series is formed from increments of a family-wise scaling process defined on the positive real line. This family-wise scaling process is expressed as an integral of a pseudo-Poisson type process. It is established that this stationary time series exhibits long-range dependence, as indicated by an autocovariance function that decays following a power law with a slowly varying component, and a spectral density that displays a power-law divergence at low frequencies. The autocovariances are not summable, indicating strong correlations over long time intervals. This framework extends the classical results on fractional Gaussian noise as well as on series driven by Poisson-type or Lévy-type noise. Additionally, it provides a versatile methodology for the spectral analysis of one-sided long-memory stochastic processes.

This paper presents some extensions of recent noncentral moderate deviation results. In the first part, the results in [Statist. Probab. Lett. 185, Paper No. 109424, 8 pp. (2022)] are generalized by considering a general Lévy process $\{S(t):t\ge 0\}$ instead of a compound Poisson process. In the second part, it is assumed that $\{S(t):t\ge 0\}$ has bounded variation and is not a subordinator; thus $\{S(t):t\ge 0\}$ can be seen as the difference of two independent nonnull subordinators. In this way, the results in [Mod. Stoch. Theory Appl. 11, 43–61] for Skellam processes are generalized.