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With the normalization used in the question, ridge regression solvesDifferentiating with respect to $\mu$ and using the centered columns $X^{T}1=0$ gives $\hat{\mu }=\overline{Y}$. The normal equation for $\beta$ issoThe push-through identity then gives the equivalent dual form

Put $Z=[X,W]$. Its ridgeless minimum-norm fit is $Z^T(Z{Z}^T{)}^{+}Y$, so its first $p$ coordinates areThe strong law of large numbers applied entrywise gives $W{W}^T\to \lambda I_n$ almost surely as $d\to \infty$. Continuity of inversion then yieldsalmost surely, where the middle equality is the push-through identity.