Abstract
We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. We apply the main result to obtain Wasserstein-2 bounds in normal approximation for sums of $m$-dependent random variables, U-statistics and subgraph counts in the Erd\H{o}s-R\'enyi random graph. We state a conjecture on Wasserstein-$p$ bounds for any positive integer $p$ and provide supporting arguments for the conjecture.
Highlights
For two probability measures μ and ν on Rd, the Wasserstein-p distance, p ≥ 1, is defined as Wp(μ, ν) = inf|x − y|pdπ(x, y) p, π∈Γ(μ,ν) where Γ(μ, ν) is the space of all probability measures on Rd × Rd with μ and ν as marginals and | · | denotes the Euclidean norm
Theorem 2.1, provides a Wasserstein-2 bound in normal approximation under local dependence, which is a generalization of independence
To prove our main result, we follow the approach of Rio [12], who used the asymptotic expansion of Barbour [1] and a Poisson-like approximation to obtain a Wasserstein-2 bound in normal approximation for sums of independent random variables
Summary
For two probability measures μ and ν on Rd, the Wasserstein-p distance, p ≥ 1, is defined as. Theorem 2.1, provides a Wasserstein-2 bound in normal approximation under local dependence, which is a generalization of independence. To prove our main result, we follow the approach of Rio [12], who used the asymptotic expansion of Barbour [1] and a Poisson-like approximation to obtain a Wasserstein-2 bound in normal approximation for sums of independent random variables. We use the triangle inequality and known Wasserstein-2 bounds in normal approximation for sums of i.i.d. random variables to prove our main result. This approach enables us to potentially bound the Wasserstein-p distance for any positive integer p. We use C to denote positive constants independent of all other parameters, possibly different from line to line
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