Let $V_1,...,V_n$ be i.i.d. the d-variate random vectors, each with a density $f(V,\beta_0)$ that satisfies the Regularity Conditions. Let $L(\beta)$ be the log-likelihood function of the observations $V_1,...,V_n$ and let $Q(\beta)$ be the penalized likelihood function $L(\beta)-n\Sigma_{j=1}^dp_{\lambda_n}(|\beta_j|)$, where $p_{\lambda_n}(|·|)=\lambda_np(|·|)$ is a penalty function and $\lambda_n>0$ is a tuning parameter.
Prove: if max$\{|p''_{\lambda_n}(\beta_{j0})|:\beta_{j0}\ne0\}\rightarrow0$,then there exists a local maximizer $\hat{\beta}$ of $Q(\beta)$ such that $\lVert\hat{\beta}-\beta_0\rVert=O_p(n^{-1/2}+a_n)$,where $a_n=max\{|p'_{\lambda_n}(|\beta_{j0}|)|:\beta_{j0}\ne0\}.$
I tried Taylor's formula, but I have no idea about the relation between $\hat{\beta}$ and $\beta_0$. When $\hat{\beta}$ is consistent for $\beta_0$, it seems easier to solve, but can that be proved?