Datasets:

zai-org
/

RPC-Bench

@@ -364,7 +364,7 @@ $$
 We can also do the same for $\begin{array} { r } { \int _ { 0 } ^ { \infty } \left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z ^ { \prime } \right) \right) ^ { m } d t } \end{array}$ .
-Because $G _ { i } ^ { * } ( h , z ) < G _ { i } ^ { * } ( h , z ^ { \prime } ) \forall h \in ( 0 , \operatorname* { m i n } ( \delta , \delta ^ { \prime } ) ) ,$ , $G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ) < G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ^ { \prime } ) \ \forall t \in ( 0 , \alpha )$ . It follows that $\begin{array} { r l r } { 1 ~ - ~ G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ) } & { { } > } & { 1 ~ - ~ G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ^ { \prime } ) } \end{array}$ and $\left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z \right) \right) ^ { m } \quad >$ 1 − G∗i (κtd/2, z0)m $\big ( 1 - G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ^ { \prime } ) \big ) ^ { m } \qquad \forall t \qquad \in \qquad ( 0 , \alpha ) .$ . So, $\begin{array} { r l r } { \int _ { 0 } ^ { \alpha } \left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z \right) \right) ^ { m } d t } & { { } } & { : } \end{array}$ > $\begin{array} { r } { \int _ { 0 } ^ { \alpha } \left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z ^ { \prime } \right) \right) ^ { m } d t } \end{array}$ .
 We now consider the second term. First, observe that $F _ { i } ^ { \theta } ( t ) ~ = ~ \operatorname* { P r } \left( \left\| { \tilde { \mathbf { x } } } _ { 1 } ^ { \theta } - \mathbf { x } _ { i } \right\| _ { 2 } ^ { 2 } \leq t \right) ~ =$ $\begin{array} { r l r } { \mathrm { P r } \left( \kappa \left\| \tilde { \mathbf { x } } _ { 1 } ^ { \theta } - \mathbf { x } _ { i } \right\| _ { 2 } ^ { d } \leq \kappa t ^ { d / 2 } \right) } & { = } & { G _ { i } ^ { \theta } \left( \kappa t ^ { d / 2 } \right) } \end{array}$ for all $t ~ \geq ~ 0$ . So, by the property of $P _ { \theta }$ , for any $\theta _ { 1 } , \theta _ { 2 }$ , there exists $\theta _ { 0 }$ such that $G _ { i } ^ { \theta _ { 0 } } \big ( \kappa t ^ { d / 2 } \big ) = F _ { i } ^ { \theta _ { 0 } } ( t ) \ge \operatorname* { m a x } \Big \{ F _ { i } ^ { \theta _ { 1 } } ( t ) , F _ { i } ^ { \theta _ { 2 } } ( t ) \Big \} = $ max $\left\{ G _ { i } ^ { \theta _ { 1 } } \big ( \kappa t ^ { d / 2 } \big ) , G _ { i } ^ { \theta _ { 2 } } \big ( \kappa t ^ { d / 2 } \big ) \right\} \forall t \geq 0$ and ∂+Gθ0i (0) = pθ0 (xi) = max {pθ1 (xi), pθ2 (xi)} = max $\left\{ \partial _ { + } G _ { i } ^ { \theta _ { 1 } } ( 0 ) , \partial _ { + } G _ { i } ^ { \theta _ { 2 } } ( 0 ) \right\}$ .

 We can also do the same for $\begin{array} { r } { \int _ { 0 } ^ { \infty } \left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z ^ { \prime } \right) \right) ^ { m } d t } \end{array}$ .
+Because $G _ { i } ^ { * } ( h , z ) < G _ { i } ^ { * } ( h , z ^ { \prime } ) \forall h \in ( 0 , \operatorname* { m i n } ( \delta , \delta ^ { \prime } ) ) ,$ , $G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ) < G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ^ { \prime } ) \ \forall t \in ( 0 , \alpha )$ . It follows that $\begin{array} { r l r } { 1 ~ - ~ G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ) } & { { } > } & { 1 ~ - ~ G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ^ { \prime } ) } \end{array}$ and $\left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z \right) \right) ^ { m } \quad >$ 1 − G∗i (κtd/2, z0)m $\big ( 1 - G _ { i } ^ { * } ( \kappa t ^ { d / 2 } , z ^ { \prime } ) \big ) ^ { m } \qquad \forall t \qquad \in \qquad ( 0 , \alpha ) .$ . So, $\begin{array} { r l r } { \int _ { 0 } ^ { \alpha } \left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z \right) \right) ^ { m } d t } & { { } } & { : } \end{array}$ > $\begin{array} { r } { \int _ { 0 } ^ { \alpha } \left( 1 - G _ { i } ^ { * } \left( \kappa t ^ { d / 2 } , z ^ { \prime } \right) \right) ^ { m } d t } \end{array}$ .
 We now consider the second term. First, observe that $F _ { i } ^ { \theta } ( t ) ~ = ~ \operatorname* { P r } \left( \left\| { \tilde { \mathbf { x } } } _ { 1 } ^ { \theta } - \mathbf { x } _ { i } \right\| _ { 2 } ^ { 2 } \leq t \right) ~ =$ $\begin{array} { r l r } { \mathrm { P r } \left( \kappa \left\| \tilde { \mathbf { x } } _ { 1 } ^ { \theta } - \mathbf { x } _ { i } \right\| _ { 2 } ^ { d } \leq \kappa t ^ { d / 2 } \right) } & { = } & { G _ { i } ^ { \theta } \left( \kappa t ^ { d / 2 } \right) } \end{array}$ for all $t ~ \geq ~ 0$ . So, by the property of $P _ { \theta }$ , for any $\theta _ { 1 } , \theta _ { 2 }$ , there exists $\theta _ { 0 }$ such that $G _ { i } ^ { \theta _ { 0 } } \big ( \kappa t ^ { d / 2 } \big ) = F _ { i } ^ { \theta _ { 0 } } ( t ) \ge \operatorname* { m a x } \Big \{ F _ { i } ^ { \theta _ { 1 } } ( t ) , F _ { i } ^ { \theta _ { 2 } } ( t ) \Big \} = $ max $\left\{ G _ { i } ^ { \theta _ { 1 } } \big ( \kappa t ^ { d / 2 } \big ) , G _ { i } ^ { \theta _ { 2 } } \big ( \kappa t ^ { d / 2 } \big ) \right\} \forall t \geq 0$ and ∂+Gθ0i (0) = pθ0 (xi) = max {pθ1 (xi), pθ2 (xi)} = max $\left\{ \partial _ { + } G _ { i } ^ { \theta _ { 1 } } ( 0 ) , \partial _ { + } G _ { i } ^ { \theta _ { 2 } } ( 0 ) \right\}$ .