Modern Stochastics: Theory and Applications

We consider an extended variant of the classical coupon collector’s problem with an infinite number of collections. An arriving coupon is placed in the rth collection, $r\ge 0$, if r is the smallest index such that the corresponding collection still does not have a coupon of this type. We derive distributional limit theorems for the number of empty spots in different collections at the time when the 0th collection was completed, as well as after some delay. We also obtain the joint limiting distribution for completion times of different collections. All main results are given in an ultimate infinite-dimensional form in the sense of distributional convergence in ${\mathbb{R}^{\infty }}$. The main tool in the proofs is convergence of specially constructed point processes.

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.

Given a compound mixed renewal process S under a probability measure P, we provide a characterization of all progressively equivalent martingale probability measures Q on the domain of P, that convert S into a compound mixed Poisson process. This result extends earlier works of Delbaen and Haezendonck, Lyberopoulos and Macheras, and the authors, and enables us to find a wide class of price processes satisfying the condition of no free lunch with vanishing risk. Implications to the ruin problem and to the computation of premium calculation principles in an arbitrage-free insurance market are also discussed.