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 ${\mathbb{R}^{\mathrm{\infty }}}$. The main tool in the proofs is convergence of specially constructed point processes.

We study dynamical Borel–Cantelli properties for the left shift on the binary symbolic space equipped with a stationary determinantal measure generated by a Toeplitz convolution kernel. For sequences of cylinder sets whose measures have a divergent sum, we give sufficient conditions ensuring infinitely many visits almost surely. Under exponential decay of the Fourier coefficients and a uniform nondegeneracy assumption, we also establish the strong Borel–Cantelli property.

This paper introduces a novel stochastic partial differential equation (SPDE) driven by space-only fractional Lévy noise ${\dot{L}_{x}^{d}}$ with $d\in (0,1/2)$, governed by a second-order elliptic operator with piecewise constant coefficients. We establish a new framework for analyzing diffusion in heterogeneous media, proving the existence, uniqueness, and equivalence of mild and weak solutions for the resulting non-Gaussian random field. A key original contribution is the explicit characterization of the covariance kernel in terms of transmission-reflection paths induced by the interface. We provide sharp estimates for the sample path regularity, proving that the solution is almost surely Hölder continuous of order $\gamma \lt d$ in space and $\beta \lt \frac{2d+1}{4}$ in time. Our results demonstrate that the fractional order of the noise robustly dictates path regularity, which remains preserved despite the singular transmission conditions and the jump structure of the driving noise.

For the heat equations driven by a stochastic measure on $[-L,L]$ with the Dirichlet boundary condition, we prove that solutions tend to the solution of the heat equations defined on $\mathbb{R}$ as $L\to \mathrm{\infty }$. The estimate of the convergence rate is obtained. For a stochastic measure, we assume the σ-additivity in probability only.

In this note, we derive a large-deviation-type scaling limit for the discretization error of a Brownian stochastic integral. Somewhat surprisingly, and to the best of our knowledge, this result has not been previously obtained in the literature.

We determine the quadratic variational cost that yields the large-deviation upper bound for the mean exit time from $(-1,1)$ in the small-noise regime for an autoregressive process driven by a continuous two-slope map f lying strictly below the diagonal on $(0,1)$. The Klebaner–Liptser reduction expresses this constant as an infimum, over all finite horizons, of quadratic path actions. Högnäs and Jung studied several nonlinear and piecewise-linear autoregressive maps and evaluated the corresponding variational problem for the two-slope family in the boundary cases in which one of the slopes vanishes; the case in which both slopes p and q are nonzero was left open. We solve this remaining case over the full admissible parameter range, including the finite-horizon attainment question, and give an exact parameter classification of the cost. We also settle the remaining boundary regimes $q=1$ and $p=0,\hspace{2.5pt}q\gt 1$, as well as the limiting case $f(1)=1$. Together with the earlier boundary formulas, this completes the exact variational analysis of continuous two-slope maps satisfying $f(x)\lt x$ for $0\lt x\lt 1$.