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For fixed coefficient dimension and $Y=X{\beta }_0+\epsilon$, the ordinary least squares error satisfiesConsequently convergence in probability of $X^{\mathsf{T}}X/n$ to a positive-definite matrix $Q$, together with $X^{\mathsf{T}}\epsilon /n\to 0$ in probability, proves statistical consistency by continuity of matrix inversion.

For independent and identically distributed random-design pairs, finite second moments and $\mathbb{E}[x_i{\epsilon }_i]=0$ imply these limits by the law of large numbers, provided $Q=\mathbb{E}[x_i{x}_i^{\mathsf{T}}]$ is nonsingular. Conditional mean zero, $\mathbb{E}[{\epsilon }_i\mid x_i]=0$, is a familiar sufficient exogeneity condition. For a deterministic design matrix, the vanishing normalized design-error product must instead be justified from its errors and weights.

Finite moments and a nonsingular design alone are insufficient. In an intercept-only model with errors identically one, the estimated intercept is always ${\beta }_0+1$. The missing zero-expected value condition changes the limiting target.