We fill two gaps in the literature on central limit theorems. First we state and prove a generalization of the Cramér-Wold device which is useful for establishing multivariate central limit theorems without the need for assuming the existence of a limiting covariance matrix. Second we extend and provide a detailed proof of a very useful result for establishing univariate central limit theorems.

Keywords: Central limit theorem, Cramér-Wold device, random field.