We want to show that for any given \(\upvarepsilon >0\), there exist a large constant C such that

$$\text\left\}=}}_^,}}_^\right)}^: \Vert }\Vert =C}}Q\left(}}_+\frac}}}_,}}_+\frac}}}_\right)<Q\left(}}_,}}_\right) \right\}\ge 1-\upvarepsilon .$$

This implies with probability \(1-\upvarepsilon \) that there exists a local maximizer of \(Q\left(}}_,}}_\right)\) in the ball \(\left\}}_+\frac}}}_,}}_+\frac}}}_: \Vert }}_^,}}_^\right)}^\Vert \le C\right\}\). Hence, there exists a local maximizer such that

$$\Vert }}}_-}}_\Vert =_\left(\frac}\right), \Vert }}}_-}}_\Vert =_\left(\frac}\right).$$

We have,

$$ \begin D\left( _ ,\user2_ } \right) = ~ & Q\left( _ + \frac}\user2_ ,\user2_ + \frac}\user2_ } \right) - ~Q\left( _ ,\user2_ } \right) \\ } = & \left[ _ + \frac}\user2_ ,\user2_ + \frac}\user2_ } \right)~ - L\left( _ ,\user2_ } \right)} \right] \\ & \; - \lambda _ \mathop \sum \limits_}^ \left\} + \frac}u_} } \right)^ - w_} } \right\} \\ ~ & \; - \lambda _ \mathop \sum \limits_}^ \left\} + \frac}u_} } \right)^ - \beta _} } \right\} \\ = & \left( } \right) - ~\left( } \right). \\ \end $$

Now, using the Taylor series approximation.

$$ \begin \left( } \right) = & ~\left[ + \frac}\user2_ ,\beta _ + \frac}\user2_ } \right)~ - L\left( ,\beta _ } \right)} \right] \\ = & L\left( ,\beta _ } \right) + \frac}\Delta _ }} L\left( ,\beta _ } \right)~\user2_ + \frac}\Delta _ }} L\left( ,\beta _ } \right)~\user2_ \\ & \; + ~\frac\user2_^ \left( } \right)\Delta _ }}^ L\left( ,\beta _ } \right)~\user2_ + \frac\user2_^ \left( } \right)\Delta _ }}^ L\left( ,\beta _ } \right)~\user2_ \\ & \; + ~\user2_^ \left( } \right)\Delta _ \beta _ }}^ L\left( ,\beta _ } \right)~\user2_ - L\left( ,\beta _ } \right) \\ ~ = & ~\frac}\Delta _ }} L\left( ,\beta _ } \right)~\user2_ + \frac}\Delta _ }} L\left( ,\beta _ } \right)~\user2_ \\ & \; + \frac\user2_^ \left( } \right)\Delta _ }}^ L\left( ,\beta _ } \right)~\user2_ + \frac\user2_^ \left( } \right)\Delta _ }}^ L\left( ,\beta _ } \right)~\user2_ \\ & \; + ~\user2_^ \left( } \right)\Delta _ \beta _ }}^ L\left( ,\beta _ } \right)~\user2_ . \\ \end $$

Since,

$$ \begin \frac}\Delta _ }} L\left( ,\beta _ } \right) = O_ \left( 1 \right) \hfill \\ \frac}\Delta _ }} L\left( ,\beta _ } \right) = O_ \left( 1 \right) \hfill \\ - \frac\Delta _ }}^ L\left( ,\beta _ } \right) \to ^ ~\user2_ }} \left( ,\beta _ } \right) \hfill \\ - \frac\Delta _ \beta _ }}^ L\left( ,\beta _ } \right) \to ^ ~\user2_ \beta _ }} \left( ,\beta _ } \right) \hfill \\ - \frac\Delta _ }}^ L\left( ,\beta _ } \right) \to ^ ~\user2_ }} \left( ,\beta _ } \right). \hfill \\ \end $$

Therefore,

\( \begin G_ = & O_ \left( 1 \right)\user2_ + O_ \left( 1 \right)\user2_ - \frac\user2_^ \user2_ }} \left( ,\beta _ } \right)\user2_ \\ & - \frac\user2_^ \user2_ \beta _ }} \left( ,\beta _ } \right)\user2_ ~ - \frac\user2_^ \user2_ }} \left( ,\beta _ } \right)\user2_ } \\ = & G_} + ~G_} + }G_} + G_} + ~G_} \\ \end \)

Here, \(_+ _\) is dominated by \(_+_+ _\) for sufficiently large \(C.\)

Expanding the function \(\frac}\right)}}}\) into a Taylor series about \(}}_\) and evaluation it at \(\widehat}}\), we get

$$ \begin \frac \right)}}}}\left. \right|_ = \user2}}} = \frac \right)}}}}\left. \right|_ = \user2}}} - \user2_}^ \left( }} \right) \hfill \\ \Rightarrow ~0 = ~\frac \right)}}}}\left. \right|_ = \user2}}} - \user2_}^ \left( }} \right) \hfill \\ \Rightarrow 0 = \frac_ } \right)}}}} - \user2_}^ \left( _ } \right) + \frac \left( _ } \right)}}\partial \user2^ }}\left( } - \user2_ } \right) - \user2_}^} \left( _ } \right)\left( } - \user2_ } \right) \hfill \\ \Rightarrow \frac_ } \right)}}}} - \user2_}^ \left( _ } \right) = \left[ \left( _ } \right)}}\partial \user2^ }} + \user2_}^} \left( _ } \right)} \right]\left( } - \user2_ } \right) \hfill \\ \Rightarrow \frac}\frac_ } \right)}}}} - \frac}\user2_}^ \left( _ } \right) \hfill \\ = \sqrt n \left( } - \user2_ } \right)\left[ \frac \left( _ } \right)}}\partial \user2^ }} + \frac\user2_}^} \left( _ } \right)} \right], \hfill \\ \end $$

Since \(E\left(\frac}}_\right)}}}\right)=0\), by central limit theorem (CLT)

$$\frac}\frac}}_\right)}}}^N\left(0,I\left(}}_\right)\right),$$

By the law of large numbers,

$$-\frac\frac^\left(}}_\right)}}\partial }}^}^E\left(-\frac\frac^\left(}}_\right)}}\partial }}^}\right).$$

That is

$$-\frac\frac^\left(}}_\right)}}\partial }}^}^I\left(}}_\right).$$

The penalty terms \(O\left(\frac\right)\), \(O\left(\frac}\right)\) are negligible asymptotically in the stochastic term.

Then,

$$ \begin \sqrt n \left( } - \user2_ } \right)\left[ _ } \right) + \frac\user2_}^} \left( _ } \right)} \right] + \frac}\user2_}^ \left( _ } \right)~ = \frac}\frac_ } \right)}}}} \hfill \\ \Rightarrow \sqrt n \left( _ } \right) + \frac\user2_}^} \left( _ } \right)} \right)~ \hfill \\ \left[ } - \user2_ } \right)~ + \left( _ } \right) + \frac\user2_}^} \left( _ } \right)} \right)^} \frac\user2_}^ \left( _ } \right)~} \right]~ = ~\frac}\frac_ } \right)}}}} \hfill \\ \Rightarrow \sqrt n ~\left[ } - \user2_ } \right)~ + \left( _ } \right) + \frac\user2_}^} \left( _ } \right)} \right)^} \frac\user2_}^ \left( _ } \right)~} \right]~ \hfill \\ = \left( _ } \right) + \frac\user2_}^} \left( _ } \right)} \right)^} ~\frac}\frac_ } \right)}}}} \hfill \\ \end $$

By the Slutsky’s theorem,

$$ \begin & \left( _ } \right) + \frac\user2_}^} \left( _ } \right)} \right)^} ~\frac}\frac_ } \right)}}}} \to ^ \left( _ } \right)} \right)^} N\left( _ } \right)} \right) \\ & = N\left( _ } \right)^} } \right), \\ \end $$

Therefore,

$$ \sqrt n ~\left[ } - \user2_ } \right)~ + \left( _ } \right) + \frac\user2_}^} \left( _ } \right)} \right)^} \frac\user2_}^ \left( _ } \right)~} \right] \to ^ N\left( _ } \right)^} } \right). $$

Let \(\text=_ \sim _^\). The moment generating function (MGF) of is

$$_\left(t\right)=^} } t<\frac.$$

The cumulative generating function (CGF) is

$$_\left(t\right)=\text\left[_\left(t\right)\right]=\text\left[^}\right]=-\frac\text\left(1-2t\right).$$

The first and second derivative of CGF is

$$ \begin K_^ \left( t \right) = - \frac \cdot \frac} \cdot \left( \right) = \frac}, \hfill \\ K_^} \left( t \right) = \left( \right)\left( \right)^} \left( \right) = \frac \right)^ ~}} \cdot \hfill \\ \end $$

Let \(\widehat\) is the solution of \(_\left(\widehat\right)=y\), then

$$_\left(\widehat\right)=y\Rightarrow \frac}=y\Rightarrow \widehat= \frac$$

Now, we define

$$ \begin w = \text \left( } \right)~\sqrt y~ - K_ \left( } \right)} \right)} \hfill \\ \Rightarrow w~ = }\left( }}} \right)\sqrt , \hfill \\ \end $$

and

$$v=\widehat\sqrt^}\left(\widehat\right)}=\frac}\cdot $$

Then, the saddle-point approximation (SPA) is used to approximate the CDF of \(\text=_ \sim _^\) to CDF of standard normal distribution, which defined as

$$_\left(y\right)=\text\left(Y<y\right)=\Phi \left(w+\frac\cdot \text\left(\frac\right)\right),$$

where \(\Phi \) is the standard normal distribution function, and

$$w=\text(\widehat) \sqrty -_\left(\widehat\right)\right)}v=\widehat\sqrt^}\left(\widehat\right)}\cdot $$