irgend so ein zahlenjunkie hat geschrieben:
begin{eqnarray*}
\big(Q_{i,n}\wedge
R_{i,n}\big)&\Longrightarrow&\exists\,t\in[S_{i-1},S_i)
\big(\vert X^\epsilon_t\vert<\epsilon^\gamma\big)\\
&\Longrightarrow& begin{eqnarray*}
\big(Q_{i,n}\wedge
R_{i,n}\big)&\Longrightarrow&\exists\,t\in[S_{i-1},S_i)
\big(\vert X^\epsilon_t\vert<\epsilon^\gamma\big)\\
&\Longrightarrow&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j+1})\;\Big(\vert
X^\epsilon_t\vert&\le&\frac12\epbegin{eqnarray*}
\big(Q_{i,n}\wedge
R_{i,n}\big)&\Longrightarrow&\exists\,t\in[S_{i-1},S_i)
\big(\vert X^\epsilon_t\vert<\epsilon^\gamma\big)\\
&\Longrightarrow&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j+1})\;\Big(\vert
X^\epsilon_t\vert&\le&\frac12\epsilon^\gamma+\vert
X^\epsilon_{S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\epsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-i+1})\;\Big(\exp(S_j-t)&\ge&\frac{(j-i+1)\theta-\frac12\epsilon^\gamma}{(j-i+1)\theta+\frac32\epsilon^\gamma}\\
&\ge&1-2\theta^{-1}\epsilon^\gamma\Big)
\end{array}\\
&\Longrightarrow&
\begin{array}{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gamma\big)\\
&\le&3\theta^{-1}\epsilon^\gamma\Big).
\end{array}
\end{eqnarray*}begin{eqnarray*}
\big(Q_{i,n}\wedge
R_{i,n}\big)&\Longrightarrow&\exists\,t\in[S_{i-1},S_i)
\big(\vert X^\epsilon_t\vert<\epsilon^\gamma\big)\\
&\Longrightarrow&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j+1})\;\Big(\vert
X^\epsilon_t\vert&\le&\frac12\epsilon^\gamma+\vert
X^\epsilon_{S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\epsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-i+1})\;\Big(\exp(S_j-t)&\ge&\frac{(j-i+1)\theta-\frac12\epsilon^\gamma}{(j-i+1)\theta+\frac32\epsilon^\gamma}\\
&\ge&1-2\theta^{-1}\epsilon^\gamma\Big)
\end{array}\\
&\Longrightarrow&
\begin{array}{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gamma\big)\\
&\le&3\theta^{-1}\epsilon^\gamma\Big).
\end{array}
\end{eqnarray*}begin{eqnarray*}
\big(Q_{i,n}\wedge
R_{i,n}\big)&\Longrightarrow&\exists\,t\in[S_{i-1},S_i)
\big(\vert X^\epsilon_t\vert<\epsilon^\gamma\big)\\
&\Longrightarrow&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j+1})\;\Big(\vert
X^\epsilon_t\vert&\le&\frac12\epsilon^\gamma+\vert
X^\epsilon_{S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\epsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-i+1})\;\Big(\exp(S_j-t)&\ge&\frac{(j-i+1)\theta-\frac12\epsilon^\gamma}{(j-i+1)\theta+\frac32\epsilon^\gamma}\\
&\ge&1-2\theta^{-1}\epsilon^\gamma\Big)
\end{array}\\
&\Longrightarrow&
\begin{array}{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gamma\big)\\
&\le&3\theta^{-1}\epsilon^\gamma\Big).
\end{array}
\end{eqnarray*}begin{eqnarray*}
\big(Q_{i,n}\wedge
R_{i,n}\big)&\Longrightarrow&\exists\,t\in[S_{i-1},S_i)
\big(\vert X^\epsilon_t\vert<\epsilon^\gamma\big)\\
&\Longrightarrow&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j+1})\;\Big(\vert
X^\epsilon_t\vert&\le&\frac12\epsilon^\gamma+\vert
X^\epsilon_{S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\epsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-i+1})\;\Big(\exp(S_j-t)&\ge&\frac{(j-i+1)\theta-\frac12\epsilon^\gamma}{(j-i+1)\theta+\frac32\epsilon^\gamma}\\
&\ge&1-2\theta^{-1}\epsilon^\gamma\Big)
\end{array}\\
&\Longrightarrow&
\begin{array}{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gamma\big)\\
&\le&3\theta^{-1}\epsilon^\gamma\Big).
\end{array}
\end{eqnarray*}begin{eqnarray*}
\big(Q_{i,n}\wedge
R_{i,n}\big)&\Longrightarrow&\exists\,t\in[S_{i-1},S_i)
\big(\vert X^\epsilon_t\vert<\epsilon^\gamma\big)\\
&\Longrightarrow&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j+1})\;\Big(\vert
X^\epsilon_t\vert&\le&\frac12\epsilon^\gamma+\vert
X^\epsilon_{S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\epsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-i+1})\;\Big(\exp(S_j-t)&\ge&\frac{(j-i+1)\theta-\frac12\epsilon^\gamma}{(j-i+1)\theta+\frac32\epsilon^\gamma}\\
&\ge&1-2\theta^{-1}\epsilon^\gamma\Big)
\end{array}\\
&\Longrightarrow&
\begin{array}{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gamma\big)\\
&\le&3\theta^{-1}\epsilon^\gamma\Big).
\end{array}
\end{eqnarray*}silon^\gamma+\vert
X^\epsilon_{{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gendamma\big)\\
&\le&3\theta^{-1}\epsilon^\S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\epsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-i+1})\;\Big(\exp(S_j-t)&\ge&\frac{(j-i+1)\theta-\frac12\epsilon^\gamma}{(j-i+1)\theta+\frac32\epsilon^\gamma}\\
&\ge&1-2\theta^{-1}\epsilon^\gamma\Big)
\end{array}\\
&\Longrightarrow&
\begin{array}{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gamma\big)\\
&\le&3\theta^{-1}\epsilon^\gamma\Big).
\end{array}
\endsilon_t\vert&\le&\frac12\epsilon^\gamma+\vert
X^\epsilon_{S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\ependendendendendendendendendendendendendsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-{eqnarray*}
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j+1})\;\Big(\vert
X^\epsilon_t\vert&\le&\frac12\epsilon^\gamma+\vert
X^\epsilon_{S_j}\vert\exp(S_j-t)\\
&\le&\frac12\epsilon^\gamma+\Big(\min\{\vert X^\epsilon_s\mid
s\in[S_{i-1},S_i)\}+(j+1)\theta\Big)\exp(S_j-t)\\
&\le&\frac32\epsilon^\gamma+(j-i+1)\theta\exp(S_j-t)\Big)
\end{array{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gendamma\big)\\
&\le&3\theta^{-1}\epsilon^\}\\
&\overset{(\ref{22})}{\Longrightarrow}&
\begin{array}{rcl}
\forall\,t\in[S_j,S_{j-i+1})\;\Big(\exp(S_j-t)&\ge&\frac{(j-i+1)\theta-\frac12\epsilon^\gamma}{(j-i+1)\theta+\frac32\epsilon^\gamma}\\
&\ge&1-2\theta^{-1}\epsilon^\gamma\Big)
\end{array}\\
&\Longrightarrow&
\begin{array}{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gendamma\big)\\
&\le&3\theta^{-1}\epsilon^\gamma\Big).
\end{array}
\end{eqnarrayend*}endendendendendende{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gendamma\big)\\
&\le&3\theta^{-1}\epsilon^\ndendendendendendendendendendendendendendendendendendendende{rcl}\Big(\tau_{j+1}&\le&\ln\big(1-2\theta^{-1}\epsilon^\gendamma\big)\\
&\le&3\theta^{-1}\epsilon^\ndendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendendend
Aldeee WTF is in diesem Fred los? 