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I'm trying to find the probability limit of

$$\hat{\beta} = \left( \sum x_i x'_i + \lambda I_k \right)^{-1} \left( \sum x_i y_i \right) $$ as $n \to \infty$, and $\lambda$ is some positive constant.

I've gotten as far as as substituting $y_i = \beta x + e_i$. (Which I know isn't very far). I'm not really sure how to progress. If anyone could provide some help that be greatly appreciated.

Thanks

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