Normal approximation by Stein's method under sublinear expectations

Peng (2008) proved the Central Limit Theorem under a sublinear expectation: Let (X-i)(i >= 1) be a sequence of i.i.d random variables under a sublinear expectation (E) over cap with (E) over cap [X-1] = (E) over cap[-X-1] = 0 and (E) over cap [1X(1)vertical bar(3)] < infinity. Setting W-n := X-1+...+X-n/root n we have, for each bounded Lipschitz function phi, lim(n ->infinity) vertical bar<(E)over cap>[phi(W-n)] - N-G(phi)vertical bar = 0, where N-G is the G-normal distribution with G(a) = 1/2 (E) over cap [aX(1)(2)], a is an element of R In this paper, we shall give an estimate of the convergence rate of this CLT by Stein's method under sublinear expectations: Under the same conditions as above, there exists a constant a is an element of (0, 1) depending on (sigma) under bar and (sigma) over bar and a positive constant C-alpha,C-G depending on alpha, (sigma) under bar and (sigma) over bar such that sup&(VERBAR;phi vertical bar <= 1) vertical bar(E) over cap[phi(W-n)] - N-G(phi) vertical bar <= C-alpha,C-G (E) over cap[vertical bar X-1 vertical bar(2+alpha)]/n(alpha/2), where (sigma) over bar (2) = (E) over cap [X-1(2)], (sigma) under bar (2) = -(E) over cap [X-(2)(1)] > 0 and N-G is the G-normal distribution with G(a) = 1/2 (E) over cap [aX(1)(2)] = 1/2 ((sigma) over bar (2) a(+) - (sigma) under bar (2)a(-)), a is an element of R. (C) 2019 Elsevier B.V. All rights reserved.