Pointwise convergence in probablity of lasso In the Knight and Fu's paper, in Equation 6 authors consider the pointwise convergence in probability as
$$\underset{\phi \in K}{\operatorname{sup}} | Z_n(\phi)-Z(\phi)-\sigma^2| \longrightarrow_p 0$$
Why is there extra $\sigma^2$ term? How did authors derive the term? How is it different from classical pointwise convergence $$\underset{\phi \in K}{\operatorname{sup}} | Z_n(\phi)-Z(\phi)| \longrightarrow_p 0$$
Can anyone point me to the proof or theory related to this?
Keith Knight and Wenjiang Fu. "Asymptotics for Lasso-Type Estimators." The Annals of Statistics, Vol. 28, No. 5 (Oct., 2000), pp. 1356-1378.
 A: Pointwise, $Z_n(\phi) \to Z(\phi) + \sigma^2$ because
\begin{align*}
Z_n(\phi) &= \frac 1 n (Y - X\phi)^T(Y - X\phi) + \frac {\lambda_n} n \sum_p |\phi_p|^\gamma \\ &= \underbrace{\frac 1 n (Y - X \hat \beta)^T(Y - X\hat \beta)}_{\to \sigma^2} \\ & \qquad + 
\underbrace{\frac 1 n (X\hat \beta - X\phi)^T(X\hat \beta - X\phi) + \frac{\lambda_n}{n} \sum_p |\phi_p|^\gamma}_{\to Z(\phi)}.
\end{align*}
So, the point of the proof is that this convergence is uniform on compact sets rather than just holding pointwise. Note that
$$
\frac 1 n (X\hat \beta - X\phi)^T(X\hat \beta - X\phi) = (\hat \beta - \phi) ^T\left[\frac 1 n X^TX\right](\hat \beta - \phi) 
$$
with $\frac 1 n X^T X \to C$ and $\hat \beta \to \beta$  (all convergence is in probability, and I'm being a little sloppy in ignoring dependence of some things on $n$). 
A: The $\sigma^2$ term is introduced at the bottom of p1357, as the dispersion parameter of the asymptotic variance of the estimator $\hat \beta^{(0)}_n$. Equation 6 is a sufficient condition for $Z_n(\phi)$ to converge pointwise to $Z(\phi) + \sigma^2$.
