We establish strong consistency of the least squares estimates in multiple regression models discarding the usual assumption of the errors having null mean value. Thus, we required them to be i.i.d. with absolute moment of order r, 0<r1. Only moderately restrictive conditions are imposed on the model matrix. In our treatment, we use an extension of the Marcinkiewicz–Zygmund strong law to overcome the errors mean value not being defined. In this way, we get a unified treatment for the case of i.i.d. errors extending the results of some previous papers.

The main purpose of this paper is to obtain strong laws of large numbers for arrays or weighted sums of random variables under a scenario of dependence. Namely, for triangular arrays $\{X_{n,j},1⩽j⩽n,n⩾1\}$ of row-wise extended negatively dependent random variables weakly mean dominated by a random variable $X\in {\mathcal{L}}_1$ and sequences $\left\{b_n\right\}$ of positive constants, conditions are given to ensure $\sum _{j=1}^n\left(X_{n,j}-\mathbb{E}{X}_{n,j}\right)/b_n\stackrel{\text{a.s.}}{⟶}0$. Our statements allow us to establish strong consistency of general nonparametric estimates in a nonparametric regression model having fixed design points.