Improve prop2 proof

index.qmd

@@ -1422,25 +1422,25 @@ $$\widehat{\mathcal{R}}_{t,\min} = 2\overline{\widehat{\mathcal{R}}}^{\text{QL}}
 First, rewrite $QL_p$ using the check function:
 
 \begin{align}
-  \rho_p(z) = z(1(0 < z) - p) \label{eq:check}
+  \rho_p(z) = z(\mathbb{1}(0 < z) - p) \label{eq:check}
 \end{align}
 
 \begin{align}
-  QL_p(x, y) &= (\mathbf1(y < x) - p)(x - y) \\
+  QL_p(x, y) &= (\mathbb{1}(y < x) - p)(x - y) \\
   &= \rho_p(x-y)
 \end{align}
 
 Now we can express the quantile risk as:
 
 \begin{align}
-  \mathcal{Q}_p(x) = E[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy
+  \mathcal{Q}_p(x) = \mathbb{E}[\rho_p(x-y)] = ∫ \rho_p(x-y)f(y)dy
 \end{align}
 
 This integral form is where the convolution becomes apparent. A convolution of functions is defined as:
 
 \begin{align}
   (g * h)(x) &= ∫ g(z)h(x - z)dz \\
-  &= ∫ h(x - z)g(z)dz
+  &= ∫ g(x - z)h(z)dz
 \end{align}
 
 They are commutative.
@@ -1477,7 +1477,7 @@ To find $\mathcal{Q}''_p(x)$ we rewrite \eqref{eq:check}:
 The function $\rho'_p(z)$ jumps from $-p$ to $1-p$ at $0$. So:
 
 \begin{align}
-  \rho''_p(x) = \delta_0(z) \quad \text{Dirac Delta}
+  \rho''_p(x) = \delta_0(z) \quad \text{(Dirac Delta)}
 \end{align}
 
 Now the magical part <i class="fa fa-fw fa-wand-magic-sparkles" style="color:var(--col_amber_6);"></i>: