Modern Stochastics: Theory and Applications

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 $p=2$.

Buraczewski et al. (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series ${\textstyle\sum _{k\ge 2}}\frac{{(\log k)^{\alpha }}}{{k^{1/2+s}}}{\eta _{k}}$ as $s\to 0+$, where $\alpha \gt -1/2$ and ${\eta _{1}},{\eta _{2}},\dots $ are independent identically distributed random variables with zero mean and finite variance. A FLT and a LIL are proved in a boundary case $\alpha =-1/2$. The boundary case is more demanding technically than the case $\alpha \gt -1/2$. A FLT and a LIL for ${\textstyle\sum _{p}}\frac{{\eta _{p}}}{{p^{1/2+s}}}$ as $s\to 0+$, where the sum is taken over the prime numbers, are stated as the conjectures.

In this paper, the distribution function and the generating function of $\varphi (u+1)$ are set up. We assume that $u\in \mathbb{N}\cup \{0\}$, $\kappa \in \mathbb{N}$, the random walk $\{{\textstyle\sum _{i=1}^{n}}{X_{i}},\hspace{0.1667em}n\in \mathbb{N}\}$ involves $N\in \mathbb{N}$ periodically occurring distributions, and the integer-valued and nonnegative random variables ${X_{1}},{X_{2}},\dots $ are independent. This research generalizes two recent works where $\{\kappa =1,N\in \mathbb{N}\}$ and $\{\kappa \in \mathbb{N},N=1\}$ were considered respectively. The provided sequence of sums $\{{\textstyle\sum _{i=1}^{n}}({X_{i}}-\kappa ),\hspace{0.1667em}n\in \mathbb{N}\}$ generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability $1-\varphi (u)$ or survival probability $\varphi (u)$. The obtained theoretical statements are verified in several computational examples where the values of the survival probability $\varphi (u)$ and its generating function are provided when $\{\kappa =2,\hspace{0.1667em}N=2\}$, $\{\kappa =3,\hspace{0.1667em}N=2\}$, $\{\kappa =5,\hspace{0.1667em}N=10\}$ and ${X_{i}}$ adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.