Modern Stochastics: Theory and Applications

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.

The problem of estimating the drift parameter is considered for an Ornstein–Uhlenbeck-type process driven by a tempered fractional Brownian motion (tfBm) or tempered fractional Brownian motion of the second kind (tfBmII). Unlike most existing studies, which assume continuous-time observations, a more realistic setting of discrete-time data is in focus. The strong consistency of a discretized least squares estimator is established under an asymptotic regime where the observation interval tends to zero while the total time horizon increases. A key step in the analysis involves deriving almost sure upper bounds for the increments of both tfBm and tfBmII.

The first-return time is the time that it takes a random walker to go back to the initial position for the first time. In this paper, the first-return time is studied when random walkers perform fractional kinetics, specifically fractional diffusion, that is modelled within the framework of the continuous-time random walk on homogeneous space in the uncoupled formulation with Mittag-Leffler distributed waiting-times. Both the Markovian and non-Markovian settings are considered, as well as any kind of symmetric jump-size distributions, namely with finite or infinite variance. It is shown that the first-return time density is indeed independent of the jump-size distribution when it is symmetric, and therefore it is affected only by the waiting-time distribution that embodies the memory of the process. The analysis is performed in two cases: first jump then wait and first wait then jump, and several exact results are provided, including the relation between results in the Markovian and non-Markovian settings and the difference between the two cases.