Approximation to stable law by the Lindeberg principle

Abstract

By Lindeberg principle, we develop in this paper an approximation to one dimensional (possibly) asymmetric α-stable distributions with α ∈ (0, 2) in smooth Wasserstein distance, which implies the stable central limit theorem. Our main tools are Taylor-like expansion and Dynkin's formula of stable process.

Approximation to stochastic variance reduced gradient Langevin dynamics by stochastic delay differential equations

“…= 0, and the equation has a unique solution: 16) then (C.6) and (C.9) imply u ∈ L 2 loc ([0, ∞) × (Ω, F , P); R d ). Since ∇ v X x r = J x 0,r v and J x 0,r J x r,t = J x 0,t , for all 0 r t, we have…”

“…Recall (C. 16) and (C.18), it is easy to see that D U 2 I x v 1 (t) can be computed by (C.14) as Let v 1 , v 2 , v 3 ∈ R d , and define u i and U i as (C.16) and (C.12), respectively, for i = 1, 2, 3. From (C.8), we can similarly define D U 3 ∇ v 2 ∇ v 1 X x s , which satisfies the following equation: for s ∈ [0, t],…”

“…Furthermore, let v 1 , v 2 , v 3 ∈ R d , and define u i and U i as (C. 16) and (C.12), respectively, for i = 1, 2, 3. From (C.19), we can similarly define…”

“…Thirdly, our method is a refined Lindeberg principle developed in [14], it seems that this method has not been reported in the study of stochastic algorithms. The classical Lindeberg principle provides an elegant way to prove the normal CLT, and has been applied to many other research topics, see [16,13,55,11,17,44,37].…”

A monotone scheme for nonlinear partial integro-differential equations with the convergence rate of $α$-stable limit theorem under sublinear expectation

“…On the other hand, if Θ is a singleton, (X t ) t≥0 is a classical Lévy process with triplet Θ, and X 1 is an α-stable random variable. The convergence rate of the classical α-stable central limit theorem has been studied in the Kolmogorov distance (see, e.g., [13, 15-17, 21, 24]) and in the Wasserstein-1 distance or the smooth Wasserstein distance (see, e.g., [1,10,11,23,30,38]). The first type is proved by the characteristic functions that do not exist in the sublinear framework, while the second type relies on Stein's method, which fails under the sublinear setting.…”