This paper provides an example of a linear regression model with predetermined stochastic regressors for which the sufficient condition for strong consistency of the ordinary least squares estimator by Lai & Wei (1982, Annals of Statistics) is not met. Nevertheless, the estimator is strongly consistent, as shown in a companion paper, cf. Christopeit & Massmann (2013b). This is intriguing because the Lai & Wei condition is the best currently available and is referred to as “in some sense the weakest possible”. Moreover, the example discussed in this paper arises naturally in a class of macroeconomic models with adaptive learning, the estimation of which has recently gained popularity amongst researchers and policy makers.