• Corpus ID: 251554672

Quantitative bounds in the central limit theorem for $m$-dependent random variables

@inproceedings{Janson2022QuantitativeBI,
  title={Quantitative bounds in the central limit theorem for \$m\$-dependent random variables},
  author={Svante Janson and Luca Pratelli and Pietro Rigo},
  year={2022}
}

• S. Janson, L. Pratelli, Pietro Rigo
• Published 12 August 2022
• Mathematics

. For each n ≥ 1, let X n, 1 ,...,X n,N n be real random variables and S n = P N n i =1 X n,i . Let m n ≥ 1 be an integer. Suppose ( X n, 1 ,...,X n,N n ) is m n -dependent, E ( X ni ) = 0, E ( X 2 ni ) < ∞ and σ 2 n := E ( S 2 n ) > 0 for all n and i . Then, d W (cid:16) S n σ n , Z (cid:17) ≤ 30 (cid:8) c 1 / 3 + 12 U n ( c/ 2) 1 / 2 (cid:9) for all n ≥ 1 and c > 0 , where d W is Wasserstein distance, Z a standard normal random variable and U n ( c ) = m n σ 2 n N n X i =1 E h X 2 n,i 1 (cid:8) | X n,i | > cσ n /m n (cid:9)i . Among other things, this estimate of d W (cid:0) S n /σ n , Z (cid:1) yields a similar estimate of d TV (cid:0) S n /σ n , Z (cid:1) where d TV is total variation distance.