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The present paper develops a unified approach when dealing with short- or long-range dependent processes with finite or infinite variance. We are concerned with the convergence rate in the strong law of large numbers (SLLN). Our main result is a Marcinkiewicz–Zygmund law of large numbers for 
$S_{n}(f)= \sum_{i=1}^{n}f(X_{i})$, where 
$\{X_i\}_{i\geq 1}$ is a real stationary Gaussian sequence and 
$f(\!\cdot\!)$ is a measurable function. Key technical tools in the proofs are new maximal inequalities for partial sums, which may be useful in other problems. Our results are obtained by employing truncation alongside new maximal inequalities. The result can help to differentiate the effects of long memory and heavy tails on the convergence rate for limit theorems.
We consider stationary configurations of points in Euclidean space that are marked by positive random variables called scores. The scores are allowed to depend on the relative positions of other points and outside sources of randomness. Such models have been thoroughly studied in stochastic geometry, e.g. in the context of random tessellations or random geometric graphs. It turns out that in a neighborhood of a point with an extreme score it is possible to rescale positions and scores of nearby points to obtain a limiting point process, which we call the tail configuration. Under some assumptions on dependence between scores, this local limit determines the global asymptotics for extreme scores within increasing windows in 
$\mathbb{R}^d$. The main result establishes the convergence of rescaled positions and clusters of high scores to a Poisson cluster process, quantifying the idea of the Poisson clumping heuristic by Aldous (1989, in the point process setting). In contrast to the existing results, our framework allows for explicit calculation of essentially all extremal quantities related to the limiting behavior of extremes. We apply our results to models based on (marked) Poisson processes where the scores depend on the distance to the kth nearest neighbor and where scores are allowed to propagate through a random network of points depending on their locations.
We study convergence rates, in mean, for the Hausdorff metric between a finite set of stationary random variables and their common support, which is supposed to be a compact subset of 
$\mathbb{R}^d$. We propose two different approaches for this study. The first is based on the notion of a minimal index. This notion is introduced in this paper. It is in the spirit of the extremal index, which is much used in extreme value theory. The second approach is based on a 
$\beta$-mixing condition together with a local-type dependence assumption. More precisely, all our results concern stationary 
$\beta$-mixing sequences satisfying a tail condition, known as the (a, b)-standard assumption, together with a local-type dependence condition or stationary sequences satisfying the (a, b)-standard assumption and having a positive minimal index. We prove that the optimal rates of the independent and identically distributed setting can be reached. We apply our results to stationary Markov chains on a ball, or to a class of Markov chains on a circle or on a torus. We study with simulations the particular examples of a Möbius Markov chain on the unit circle and of a Markov chain on the unit square wrapped on a torus.
$\chi^2$-random fields on two-point homogeneous spaces
We investigate geometric properties of invariant spatio-temporal random fields 
$X\colon\mathbb M^d\times \mathbb R\to \mathbb R$ defined on a compact two-point homogeneous space 
$\mathbb M^d$ in any dimension 
$d\ge 2$, and evolving over time. In particular, we focus on chi-squared-distributed random fields, and study the large-time behavior (as 
$T\to +\infty$) of the average on [0,T] of the volume of the excursion set on the manifold, i.e. of 
$\lbrace X(\cdot, t)\ge u\rbrace$ (for any 
$u >0$). The Fourier components of X may have short or long memory in time, i.e. integrable or non-integrable temporal covariance functions. Our argument follows the approach developed in Marinucci et al. (2021) and allows us to extend their results for invariant spatio-temporal Gaussian fields on the two-dimensional unit sphere to the case of chi-squared distributed fields on two-point homogeneous spaces in any dimension. We find that both the asymptotic variance and limiting distribution, as 
$T\to +\infty$, of the average empirical volume turn out to be non-universal, depending on the memory parameters of the field X.
For a spectrally negative Lévy process X, consider 
$g_t$ and its infinitesimal generator. Moreover, with 
$t\geq 0$, the last time X is below the level zero before time 
$\{(g_t,t, X_t), t\geq 0 \}$ the length of a current positive excursion, we derive a general formula that allows us to calculate a functional of the whole path of 
$U_t\,:\!=\,t-g_t$. We use a perturbation method for Lévy processes to derive an Itô formula for the three-dimensional process 
$ (U, X)=\{(U_t, X_t),t\geq 0\}$ in terms of the positive and negative excursions of the process X. As a corollary, we find the joint Laplace transform of 
$(U_{\mathbf{e}_q}, X_{\mathbf{e}_q})$, where 
$\mathbf{e}_q$ is an independent exponential time, and the q-potential measure of the process (U, X). Furthermore, using the results mentioned above, we find a solution to a general optimal stopping problem depending on (U, X) with an application in corporate bankruptcy. Lastly, we establish a link between the optimal prediction of 
$g_{\infty}$ and optimal stopping problems in terms of (U, X) as per Baurdoux, E. J. and Pedraza, J. M., 
$L_p$ optimal prediction of the last zero of a spectrally negative Lévy process, Annals of Applied Probability, 34 (2024), 1350–1402.
We show that 
$\alpha $-stable Lévy motions can be simulated by any ergodic and aperiodic probability-preserving transformation. Namely we show that: for 
$0<\alpha <1$ and every 
$\alpha $-stable Lévy motion 
${\mathbb {W}}$, there exists a function f whose partial sum process converges in distribution to 
${\mathbb {W}}$; for 
$1\leq \alpha <2$ and every symmetric 
$\alpha $-stable Lévy motion, there exists a function f whose partial sum process converges in distribution to 
${\mathbb {W}}$; for 
$1< \alpha <2$ and every 
$-1\leq \beta \leq 1$ there exists a function f whose associated time series is in the classical domain of attraction of an 
$S_\alpha (\ln (2), \beta ,0)$ random variable.
The payoff in the Chow–Robbins coin-tossing game is the proportion of heads when you stop. Stopping to maximize expectation was addressed by Chow and Robbins (1965), who proved there exist integers 
${k_n}$ such that it is optimal to stop at n tosses when heads minus tails is 
${k_n}$. Finding 
${k_n}$ was unsolved except for finitely many cases by computer. We prove an 
$o(n^{-1/4})$ estimate of the stopping boundary of Dvoretsky (1967), which then proves 
${k_n} = \left\lceil {\alpha \sqrt n \,\, - 1/2\,\, + \,\,\frac{{\left( { - 2\zeta (\! -1/2)} \right)\sqrt \alpha }}{{\sqrt \pi }}{n^{ - 1/4}}} \right\rceil $ except for n in a set of density asymptotic to 0, at a power law rate. Here, 
$\alpha$ is the Shepp–Walker constant from the Brownian motion analog, and 
$\zeta$ is Riemann’s zeta function. An 
$n^{ - 1/4}$ dependence was conjectured by Christensen and Fischer (2022). Our proof uses moments involving Catalan and Shapiro Catalan triangle numbers which appear in a tree resulting from backward induction, and a generalized backward induction principle. It was motivated by an idea of Häggström and Wästlund (2013) to use backward induction of upper and lower Value bounds from a horizon, which they used numerically to settle a few cases. Christensen and Fischer, with much better bounds, settled many more cases. We use Skorohod’s embedding to get simple upper and lower bounds from the Brownian analog; our upper bound is the one found by Christensen and Fischer in another way. We use them first for yet many more examples and a conjecture, then algebraically in the tree, with feedback to get much sharper Value bounds near the border, and analytic results. Also, we give a formula that gives the exact optimal stop rule for all n up to about a third of a billion; it uses the analytic result plus terms arrived at empirically.
Consider nested subdivisions of a bounded real set into intervals defining the digits 
$X_1,X_2,\ldots$ of a random variable X with a probability density function f. If f is almost everywhere lower semi-continuous, there is a non-negative integer-valued random variable N such that the distribution of 
$R=(X_{N+1},X_{N+2},\ldots)$ conditioned on 
$S=(X_1,\ldots,X_N)$ does not depend on f. If also the lengths of the intervals exhibit a Markovian structure, 
$R\mid S$ becomes a Markov chain of a certain order 
$s\ge0$. If 
$s=0$ then 
$X_{N+1},X_{N+2},\ldots$ are independent and identically distributed with a known distribution. When 
$s>0$ and the Markov chain is uniformly geometric ergodic, there is a random time M such that the chain after time 
$\max\{N,s\}+M-s$ is stationary and M follows a simple known distribution.
We introduce the exponentially preferential recursive tree and study some properties related to the degree profile of nodes in the tree. The definition of the tree involves a radix 
$a\gt 0$. In a tree of size 
$n$ (nodes), the nodes are labeled with the numbers 
$1,2, \ldots ,n$. The node labeled 
$i$ attracts the future entrant 
$n+1$ with probability proportional to 
$a^i$.
We dedicate an early section for algorithms to generate and visualize the trees in different regimes. We study the asymptotic distribution of the outdegree of node 
$i$, as 
$n\to \infty$, and find three regimes according to whether 
$0 \lt a \lt 1$ (subcritical regime), 
$a=1$ (critical regime), or 
$a\gt 1$ (supercritical regime). Within any regime, there are also phases depending on a delicate interplay between 
$i$ and 
$n$, ramifying the asymptotic distribution within the regime into “early,” “intermediate” and “late” phases. In certain phases of certain regimes, we find asymptotic Gaussian laws. In certain phases of some other regimes, small oscillations in the asymototic laws are detected by the Poisson approximation techniques.
We consider a superprocess 
$\{X_t\colon t\geq 0\}$ in a random environment described by a Gaussian field 
$\{W(t,x)\colon t\geq 0,x\in \mathbb{R}^d\}$. First, we set up a representation of 
$\mathbb{E}[\langle g, X_t\rangle\mathrm{e}^{-\langle \,f,X_t\rangle }\mid\sigma(W)\vee\sigma(X_r,0\leq r\leq s)]$ for 
$0\leq s < t$ and some functions f,g, which generalizes the result in Mytnik and Xiong (2007, Theorem 2.15). Next, we give a uniform upper bound for the conditional log-Laplace equation with unbounded initial values. We then use this to establish the corresponding conditional entrance law. Finally, the excursion representation of 
$\{X_t\colon t\geq 0\}$ is given.
Considering a double-indexed array 
$(Y_{n,i:\,n\ge 1,i\ge 1})$ of non-negative regularly varying random variables, we study the random-length weighted sums and maxima from its ‘row’ sequences. These sums and maxima may have the same tail and extremal indices (Markovich and Rodionov 2020). The main constraints of the latter results are that there exists a unique series in a scheme of series with the minimum tail index and the tail of the term number is lighter than the tail of the terms. Here, a bounded random number of series are allowed to have the minimum tail index and the tail of the term number may be heavier than the tail of the terms. We derive the tail and extremal indices of the weighted non-stationary random-length sequences under a broader set of conditions than in Markovich and Rodionov (2020). We provide examples of random sequences for which the assumptions are valid. Perspectives in adopting the results in different application areas are formulated.
Let 
$(X_t)_{t \geqslant 0}$ be the solution of the stochastic differential equation where 
\[\mathrm{d} X_t = b(X_t) \,\mathrm{d} t+A\,\mathrm{d} Z_t, \quad X_{0}=x,\]
$b\,:\, \mathbb{R}^d \rightarrow \mathbb{R}^d$ is a Lipschitz-continuous function, 
$A \in \mathbb{R}^{d \times d}$ is a positive-definite matrix, 
$(Z_t)_{t\geqslant 0}$ is a d-dimensional rotationally symmetric 
$\alpha$-stable Lévy process with 
$\alpha \in (1,2)$ and 
$x\in\mathbb{R}^{d}$. We use two Euler–Maruyama schemes with decreasing step sizes 
$\Gamma = (\gamma_n)_{n\in \mathbb{N}}$ to approximate the invariant measure of 
$(X_t)_{t \geqslant 0}$: one uses independent and identically distributed 
$\alpha$-stable random variables as innovations, and the other employs independent and identically distributed Pareto random variables. We study the convergence rates of these two approximation schemes in the Wasserstein-1 distance. For the first scheme, under the assumption that the function b is Lipschitz and satisfies a certain dissipation condition, we demonstrate a convergence rate of 
$\gamma^{\frac{1}{\alpha}}_n$. This convergence rate can be improved to 
$\gamma^{1+\frac {1}{\alpha}-\frac{1}{\kappa}}_n$ for any 
$\kappa \in [1,\alpha)$, provided b has the additional regularity of bounded second-order directional derivatives. For the second scheme, where the function b is assumed to be twice continuously differentiable, we establish a convergence rate of 
$\gamma^{\frac{2-\alpha}{\alpha}}_n$; moreover, we show that this rate is optimal for the one-dimensional stable Ornstein–Uhlenbeck process. Our theorems indicate that the recent significant result of [34] concerning the unadjusted Langevin algorithm with additive innovations can be extended to stochastic differential equations driven by an 
$\alpha$-stable Lévy process and that the corresponding convergence rate exhibits similar behaviour. Compared with the result in [6], our assumptions have relaxed the second-order differentiability condition, requiring only a Lipschitz condition for the first scheme, which broadens the applicability of our approach.
