Some strong consistency results in stochastic regression

Strong consistency of the least-squares estimates in stochastic regression models is established assuming errors with variance not necessarily defined. The errors will be considered identically distributed having absolute moment of order r, 0<r⩽2 and, additionally, pairwise independent whenever r=2. It is shown that only a moderate asymptotic assumption on the stochastic regressors is sufficient to obtain strong consistency of the least-squares estimates allowing that both exponential and linear asymptotic behavior for the squared sums of the design levels can coexist. Strong consistency of the ridge estimates is also obtained for some biasing parameters using the previous assumptions on the errors.