Central limit theorem and the Slepian-Stein method

Roughly speaking, the celebrated central limit theorem says that a sum of many small independent random variables with sufficiently nice distributions approximately follows the normal (Gaussian) distribution. An important issue is the estimation of the error in the normal approximation. Numerous approaches have been suggested. In 1962, Slepian suggested elegant gradual replacement of the original sum with a Gaussian random variable. This approach is essentially equivalent to Stein’s method introduced in 1970, which arises from a different idea.

Though relatively old, Stein’s method is still a highly active field of research. This is highly due to its flexibility. In particular, Stein’s method does not only work for sums of independent random variables, but also for families with various kind of dependence structure. We shall focus on the so called local dependence.