On strong embeddings by Stein’s method

Abstract

Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee's novel use of Stein's method for {−1, +1} valued variables to a general class of discrete distributions, and provide log n rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge.Since the expectation of g(X * ) and g(U X ) agree for any g ∈ C c , with = d denoting distributional equivalence, we obtainSmoothing X by adding an independent random variable Y having the X -zero bias distribution, we obtain the following result which will be used for constructing Stein coefficients for sums. Lemma 2.4. If X is a mean zero random variable with finite non-zero variance, and Y is an independent variable with the X -zero bias distribution, thenfor all Lipschitz functions f and a.e. derivative f ′ for which these expectations exist.

One idea and two proofs of the KMT theorems

“…n S 2 n + 3. The Bernstein/Hoeffding inequality says that P{S n ≥ t} ≤ e −t 2 /2n , which can be interpreted as saying that (S n ) 2 + and (S n ) 2 − are stochastically dominated by 2nξ, where ξ is an exponential random variable with unit mean.…”

“…The second approach, due to Chatterjee [4], proves KMT-RW when the steps have symmetric Bernoulli distribution, by an approach that may be broadly described as Stein's method. Bhattacharjee and Goldstein [2] extended this method of proof to a large class of step distributions.…”

“…◮ part (3) of Lemma 21, there exists α 0 > 0 and γ < 1 such that for any n ≥ 1 and any k ≤2 3 n, any probable value t of S n , and any α ≤ α 0 , Assume that the conclusion (31) holds for any m < n in place of n (and any probable value t of S m ). Fix n ≥ 6 and a probable value t of S n .…”

Bound on the running maximum of a random walk with small drift

“…The difficulty of the original KMT proof has motivated several recent attempts at simplification and better understanding of the result, such as Bhattacharjee and Goldstein (2016), Chatterjee (2012), and Krishnapur (2020). There is another strong approximation result due to Sakhanenko (1984) which, according to p. 232 of Chatterjee (2012), "is so complex that some researchers are hesitant to use it".…”

Bounds on the running maximum of a random walk with small drift

“…The difficulty of the original KMT proof has motivated several recent attempts at simplification and better understanding of the result, such as Bhattacharjee and Goldstein [1], Chatterjee [3], and Krishnapur [8]. There is another strong approximation result due to Sakhanenko [9] which, according to p. 232 of [3], "is so complex that some researchers are hesitant to use it".…”