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Let 
$\mathbb{P}_\kappa(n)$ be the probability that n points 
$z_1,\ldots,z_n$ picked uniformly and independently in 
$\mathfrak{C}_\kappa$, a regular 
$\kappa$-gon with area 1, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we compute 
$\mathbb{P}_\kappa(n)$ up to asymptotic equivalence, as 
$n\to+\infty$, for all 
$\kappa\geq 3$, which improves on a famous result of Bárány (Ann. Prob. 27, 1999). The second purpose of this paper is to establish a limit theorem which describes the fluctuations around the limit shape of an n-tuple of points in convex position when 
$n\to+\infty$. Finally, we give an asymptotically exact algorithm for the random generation of 
$z_1,\ldots,z_n$, conditioned to be in convex position in 
$\mathfrak{C}_\kappa$.
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.
We study continuous-time Markov chains on the nonnegative integers under mild regularity conditions (in particular, the set of jump vectors is finite and both forward and backward jumps are possible). Based on the so-called flux balance equation, we derive an iterative formula for calculating stationary measures. Specifically, a stationary measure 
$\pi(x)$ evaluated at 
$x\in\mathbb{N}_0$ is represented as a linear combination of a few generating terms, similarly to the characterization of a stationary measure of a birth–death process, where there is only one generating term, 
$\pi(0)$. The coefficients of the linear combination are recursively determined in terms of the transition rates of the Markov chain. For the class of Markov chains we consider, there is always at least one stationary measure (up to a scaling constant). We give various results pertaining to uniqueness and nonuniqueness of stationary measures, and show that the dimension of the linear space of signed invariant measures is at most the number of generating terms. A minimization problem is constructed in order to compute stationary measures numerically. Moreover, a heuristic linear approximation scheme is suggested for the same purpose by first approximating the generating terms. The correctness of the linear approximation scheme is justified in some special cases. Furthermore, a decomposition of the state space into different types of states (open and closed irreducible classes, and trapping, escaping and neutral states) is presented. Applications to stochastic reaction networks are well illustrated.
Motivated by recent developments of quasi-stationary Monte Carlo methods, we investigate the stability of quasi-stationary distributions of killed Markov processes under perturbations of the generator. We first consider a general bounded self-adjoint perturbation operator, and then study a particular unbounded perturbation corresponding to truncation of the killing rate. In both scenarios, we quantify the difference between eigenfunctions of the smallest eigenvalue of the perturbed and unperturbed generators in a Hilbert space norm. As a consequence, 
$\mathcal{L}^1$-norm estimates of the difference of the resulting quasi-stationary distributions in terms of the perturbation are provided.