Modern Stochastics: Theory and Applications

This paper investigates fractional Riesz–Bessel equations with random initial conditions that exhibit either classical or cyclic long-range dependence. It studies zoom-in asymptotics for the corresponding solutions and establishes multiscaling limit theorems. It is known that for similar problems, non-degenerate multiscaling limits may not exist in general. The paper develops a kernel-smoothing approach for these equations and obtains non-degenerate limit fields under suitable normalisation and rescaling. It proves that the kernel-smoothed solutions converge weakly to Gaussian random fields, which are non-stationary in both time and space. Their stochastic integral representations and covariance functions are derived. The paper also analyses the regularity and dependence structure of the limit fields. In particular, under appropriate general assumptions on the smoothing kernel, the limits exhibit long-range dependence in time and short-range dependence in space. Numerical examples for the case of Matérn-type kernels are provided to illustrate the theoretical results.

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 this paper, we introduce a family of semi-parametric estimators for the positive extreme value index γ, parameterized in two tuning parameters. The asymptotic normality of the introduced estimators is proved. It is shown that the partial case of newly introduced estimators (a subfamily with one tuning parameter) has quite good asymptotic properties and dominates several previously introduced estimators. Small Monte-Carlo simulations are included. Also, the performance of this parameterized subfamily of estimators is illustrated for pair exchange ratio data sets.

We prove a limit theorem for paths of random walks with n steps in ${\mathbb{R}^{d}}$ as n and d both go to infinity. For this, the paths are viewed as finite metric spaces equipped with the ${\ell _{p}}$-metric for $p\in [1,\infty )$. Under the assumptions that all components of each step are uncorrelated, centered, have finite $2p$-th moments, and are identically distributed, we show that such random metric space converges in probability to a deterministic limit space with respect to the Gromov-Hausdorff distance. This result generalises earlier work by Kabluchko and Marynych [Ann. Inst. H. Poincaré Probab. Statist. 60(4): 2945–2974, 2024] for $p=2$.

Multisets are like sets, except that they can contain multiple copies of their elements. If there are ${n_{i}}$ copies of i, $1\le i\le t$, in multiset ${M_{t}}$, then there are $\left(\genfrac{}{}{0.0pt}{}{{n_{1}}+\cdots +{n_{t}}}{{n_{1}},\dots ,{n_{t}}}\right)$ possible permutations of ${M_{t}}$. Knuth showed how to factor any multiset permutation into cycles. For fixed ${n_{i}}$, $i\ge 1$, we show how to adapt the Chinese restaurant process, which generates random permutations on n elements with weighting ${\theta ^{\# \hspace{0.1667em}\mathrm{cycles}}}$, $\theta \gt 0$, sequentially for $n=1,2,\dots $, to the multiset case, where we fix the ${n_{i}}$ and build permutations on ${M_{t}}$ sequentially for $t=1,2,\dots $. The number of cycles of a multiset permutation chosen uniformly at random, i.e. $\theta =1$, has distribution given by the sum of independent negative hypergeometric distributed random variables. For all $\theta \gt 0$, and under the assumption that ${n_{i}}=O(1)$, we show a central limit theorem as $t\to \infty $ for the number of cycles.

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.