\documentstyle[12pt,fleqn]{article}
\def\argmax{\mathop{\rm argmax}}

\pagestyle{empty}
\begin{document}
\thispagestyle{empty}




%-------------------------------------------------------------------------%

\begin{eqnarray}
d&=&\{O_{1},O_{2}, \ldots, O_{T}\}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%


\begin{eqnarray}
\pi(x) &=& Pr(x_0=x)\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\mu(x\mid x') &=& Pr(x_t=x \mid  x_{t-1} =x')\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\nu(b\mid x) &=& Pr(O_t=b \mid  x_t = x)\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\lambda&=& \{\pi,\mu,\nu\}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
&& \hskip -15mm Pr(x_t\mid  O_1,\ldots,O_t,\lambda) \nonumber\\ \nonumber\\
&=&\frac{Pr(O_t \mid  O_1,\ldots,O_{t-1},x_t,\lambda) \; Pr(x_t\mid  O_1,\ldots,O_{t-1},\lambda)}{Pr(O_t \mid  O_1,\ldots,O_{t-1},\lambda)}
\nonumber\\ \nonumber\\
&=&\eta\;Pr(O_t \mid  x_t,\lambda) \; Pr(x_t\mid  O_1,\ldots,O_{t-1},\lambda)
\nonumber\\ \nonumber\\
&=&\eta\;Pr(O_t \mid  x_t,\lambda) \; \int Pr(x_t \mid  x_{t-1}, O_1,\ldots,O_{t-1},\lambda) \;Pr(x_{t-1}\mid  O_1,\ldots,O_{t-1},\lambda)\; dx_{t-1}
\nonumber\\ \nonumber\\
&=&\eta\;Pr(O_t \mid  x_t,\lambda) \; \int Pr(x_t \mid  x_{t-1},\lambda) \;Pr(x_{t-1}\mid  O_1,\ldots,O_{t-1},\lambda)\; dx_{t-1}
\nonumber\\ \nonumber\\
&=&\eta\;\nu(O_t, x_t) \; \int \mu(x_t, x_{t-1}) \;Pr(x_{t-1}\mid  O_1,\ldots,O_{t-1},\lambda)\; dx_{t-1}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
{\bf x}^{(n)} &:=& Pr(x_1,\ldots,x_t \mid  d, \lambda^{(n)})\nonumber
\end{eqnarray}
\newpage

\begin{eqnarray}
\lambda^{(n+1)} &:=& \argmax_{\lambda}Pr(\lambda^{(n)} \mid  {\bf x}^{(n)}, d)\nonumber
\end{eqnarray}

\newpage
\begin{eqnarray}
\lambda^* &:=& \argmax_{\lambda}Pr(\lambda \mid   d)\nonumber
\end{eqnarray}
\newpage


%-------------------------------------------------------------------------%

\begin{eqnarray}
\gamma_t^{(n)}(x) &=& Pr(x_t=x \mid d, \lambda^{(n)})\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\alpha_t^{(n)}(x) &=& Pr(x_t=x\mid  O_1,\ldots,O_t,\lambda^{(n)})\nonumber
\end{eqnarray}
\newpage


%-------------------------------------------------------------------------%

\begin{eqnarray}
\beta_t^{(n)}(x) &=& Pr( O_{t+1},\ldots,O_{T}\mid  x_t=x, \lambda^{(n)})\nonumber
\end{eqnarray}
\newpage



%-------------------------------------------------------------------------%

\begin{eqnarray}
\alpha_0^{(n)}(x)&=&\pi^{(n)}(x)
\label{alphaInit}\nonumber\\\nonumber\\
\alpha_t^{(n)}(x)&=& \eta\; \nu^{(n)}(O_t \mid  x)\;\int \alpha_{t-1}^{(n)}(x') \;\mu^{(n)}(x\mid  x') \;dx'
\label{eqAlpha}\nonumber
\end{eqnarray}
\newpage


%-------------------------------------------------------------------------%

\begin{eqnarray}
\beta_{T}^{(n)}(x) &=& 1
\label{betaInit}
\nonumber\\\nonumber\\
\beta_t^{(n)}(x) &=&\eta\; \int \beta_{t+1}^{(n)}(x') \;\mu^{(n)}(x'\mid  x)\;\nu^{(n)}(O_{t+1} \mid  x')\;dx'
\label{eqBeta}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%




\begin{eqnarray}
\gamma_t^{(n)}(x) &=&
Pr(x_t=x \mid d, \lambda^{(n)})\nonumber\\
&=&\eta\;\alpha_t^{(n)}(x)\;\beta_t^{(n)}(x)
\nonumber
\end{eqnarray}
\newpage



%-------------------------------------------------------------------------%

\begin{eqnarray}
\xi_t^{(n)}(x,x') &=& Pr(x_t=x , x_{t+1}=x' \mid  d, \lambda^{(n)}) \nonumber\\\nonumber\\
 &=& \eta\;\alpha_{t}^{(n)}(x)\;\mu^{(n)}(x'\mid  x)\;\nu^{(n)}(O_{t+1}\mid  x')\;\beta_{t+1}^{(n)}(x)
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\pi^{(n+1)}(x) &=& \gamma_0^{(n)}(x)
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\mu^{(n+1)}(x'\mid  x) &=& \frac{\displaystyle  \sum_{t=1}^{T-1} \xi_t^{(n)}(x,x')}{\displaystyle \sum_{t=1}^{T-1} \gamma_t^{(n)}(x)}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\nu^{(n+1)}(b\mid  x) &=&  \frac{\displaystyle  \sum_{t=1}^{T} I_{O_t=b} \; \gamma_t^{(n)}(x)}{ \displaystyle \sum_{t=1}^{T} \gamma_t^{(n)}(x)}
\label{eqNu}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
x_{n} && \mbox{is drawn according to $f$}\nonumber\nonumber\\\nonumber\\
p_{x_n} &=& N^{-1}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
x_{n} && \mbox{is drawn uniformly from ${\rm dom}(f)$}\nonumber\nonumber\\\nonumber\\
p_{x_n} &=& f(x_{n})\;\left[\sum_{i=1}^{N} f(x_{i})\right]^{-1}
\nonumber
\end{eqnarray}
\newpage


%-------------------------------------------------------------------------%

\begin{eqnarray}
x_{n} && \mbox{is drawn according to $g$}\nonumber\\
&& \mbox{for any $g$ with $f(x)>0\Longrightarrow g(x)>0$}\nonumber\\\nonumber\\
p_{x_n} &=& \frac{f(x_{n})}{g(x_{n})} \;\left[\sum_{i=1}^{N} \frac{f(x_{i})}{g(x_{i})}\right]^{-1}
\nonumber
\end{eqnarray}
\newpage


%-------------------------------------------------------------------------%

\begin{eqnarray}
\lim_{N \rightarrow \infty}\;\;\sum_{\langle x,p_x\rangle\in X} I_{x_0\leq x < x_1} \; p_x
&= &
\int_{x_0}^{x_1} f(x)\;dx\;\;\;\;\;\;\mbox{w.p. 1}
\label{eqConv}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
O_t &=&\left\{\begin{array}{ll}
(0.25+\varepsilon_{t,1}, 0.25+\varepsilon_{t,2},0.25+\varepsilon_{t,3},\ldots, 0.25+\varepsilon_{t,10}) &
\mbox{if $t$ odd} \nonumber\\\nonumber\\
(0.75+\varepsilon_{t,1},0.75+\varepsilon_{t,2},0.75+\varepsilon_{t,3}, \ldots, 0.75+\varepsilon_{t,10}) &
\mbox{if $t$ even} 
\end{array}\right.
\nonumber
\end{eqnarray}      
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
O_t &=&\left\{\begin{array}{ll}
(0.25+\varepsilon_{t,1}, 0.75+\varepsilon_{t,2},0.25+\varepsilon_{t,3}, \ldots, 0.75+\varepsilon_{t,10}) &
\mbox{if $t$ odd} \nonumber\\\nonumber\\
(0.75+\varepsilon_{t,1},0.25+\varepsilon_{t,25},0.75+\varepsilon_{t,3} \ldots, 0.25+\varepsilon_{t,10}) &
\mbox{if $t$ even} 
\end{array}\right.
\nonumber
\end{eqnarray}      

\end{document}


 END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END END

\newpage




%-------------------------------------------------------------------------%

\begin{eqnarray}
\lim _{K \rightarrow \infty} \sum _{k=1}^K c_{kx} \;b_{kx}(O)
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

 \begin{eqnarray}
 \int_Y
\int_X f(x,y) \;dx\; dy &=& \int_X \int_Y f(x,y) \;dy\; dx 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

 \begin{eqnarray}
 \nabla _Y
\int_X f(x,y) \;dx  &=& \int_X \nabla _Y f(x,y) \; dx 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
X&=&\{\langle x_{1},p_{x_1}\rangle\, \ldots,\langle x_{N},p_{x_N}\rangle \}
\;\;\;\;\mbox{with}\;\sum_{n=1}^{N}
p_{x_n}=1
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
f(x)>0 &\Longrightarrow g(x)>0
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\sum_{\langle x,p_x\rangle\in X_{\rm u}} f(x) &\leq& 
\sum_{\langle x,p_x\rangle\in X_{\rm lw}} f(x)
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
Pr\left(\;\left| \sum_{\langle x,p_x\rangle\in X} I_{x_0\leq x < x_1} \;
p_x - \int_{x_0}^{x_1} f(x)\;dx\right|\;> \varepsilon\;\right) &\leq& 2\;e^{-2\varepsilon^2 N}
\label{eqTheorem2}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\hat{\sigma}_i &=& \sum_{\langle x,p_{x}\rangle\in X}I_{x\in V_{i}}\; p_{x}
\label{eqSigma}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\hat{f}(x) &=& \frac{\hat{\sigma}_{i(x)}}{| V_{i(x)} |}\;\;\;\;\;\;\;\;\;\;(\forall x\in {\rm dom}(f))
\label{eqTreeDens}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\lim_{N \rightarrow \infty}\;\;
\int_{x_0}^{x_1} \hat{f}(x)\;dx
&= &
\int_{x_0}^{x_1} f(x)\;dx \;\;\;\;\;\;\mbox{w.p. 1}.
\label{eqTheorem3a}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

% \begin{eqnarray}
% Pr\left(\;\left|
% \int_{x_0}^{x_1} \hat{f}(x)\;dx
% - \int_{x_0}^{x_1} f(x)\;dx\right| > \varepsilon\right) &\leq & ...?
% \label{eqTheorem3a}
%\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\lim_{N \rightarrow \infty}\;\;
\hat{f}(x) &=
&f(x)\;\;\;\;\;\;\;\;\;\;(\forall x\in {\rm dom}(f))\;\;\;\;\;\;\mbox{w.p. 1}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\bar{\sigma_i} &=&\int_{V_i} f(x)\;dx.
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
Pr(|\bar{\sigma_i} - \hat{\sigma}_i| > \varepsilon )&<&
2\;e^{-2\varepsilon^2 N}.
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
2^{\frac{1}{4} \log_2 N} &=&  N^{\frac{1}{4}}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
Pr(\exists \;{\rm leaf} \;i: |\bar{\sigma_i} - \hat{\sigma}_i| > \varepsilon) &<&
2\;N^{\frac{1}{4}}\;e^{-2\varepsilon^2 N}.
\label{eq1}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
|\bar{f}(x)-\hat{f}(x)|
&=& |V_i|^{-1}\; | \bar{\sigma_i} - \hat{\sigma}_i|
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
&&|\bar{f}(x)-\hat{f}(x)|
\;\;\leq\;\; N^{-\frac{1}{4}}\; | \bar{\sigma_i} - \hat{\sigma}_i|
\nonumber\\\nonumber\\
&\Longleftrightarrow&
N^{\frac{1}{4}}\;|\bar{f}(x)-\hat{f}(x)|
\;\;\leq\;\;  | \bar{\sigma_i} - \hat{\sigma}_i|
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
Pr( |\bar{f}(x)-\hat{f}(x)| > N^{\frac{1}{4}}\varepsilon) &<& 2\;N^{\frac{1}{4}}\;e^{-2\varepsilon^2 N},
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
Pr( |\bar{f}(x)-\hat{f}(x)| > \varepsilon' )&<& 2\;N^{\frac{1}{4}}\;e^{-2\varepsilon'^2  \sqrt{N}}.
\label{eq2}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\bar{\sigma}_i& >& 0
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
n_i &=&  \sum_{\langle x,p_{x}\rangle\in X}I_{x\in V_{i}} \;\;=\;\;
N\;\hat{\sigma}_i
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
N &>& \max\left\{\frac{1}{(\bar{\sigma}_i-\varepsilon)^2}\;,\;4^{{\rm
depth}(i)} \right\}\;\;\;\;\;\;\;\;\mbox{and}\;\;\;\;\;\;\;\;\varepsilon <
\bar{\sigma}_i
\label{eqMax}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\bar{\sigma}_i \;\; > \;\; \frac{1}{\sqrt{N}} + \varepsilon
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
&&Pr( \bar{\sigma}_i - \hat{\sigma}_i  > \varepsilon) \;\;\leq\;\; e^{-2\varepsilon^2 N}\nonumber\\\nonumber\\
&\Longrightarrow&
Pr\left(\frac{1}{\sqrt{N}} + \varepsilon  - \hat{\sigma}_i  > \varepsilon\right) \;\;\leq\;\; e^{-2\varepsilon^2 N}\nonumber\\\nonumber\\
&\Longleftrightarrow&
Pr\left(\frac{1}{\sqrt{N}} > \hat{\sigma}_i\right) \;\;\leq\;\; e^{-2\varepsilon^2 N}\nonumber\\\nonumber\\
&\Longleftrightarrow&
Pr\left(\frac{1}{\sqrt{N}} > \frac{n_i}{N}\right) \;\;\leq\;\; e^{-2\varepsilon^2 N}\nonumber\\\nonumber\\
&\Longleftrightarrow&
Pr\left(\sqrt{N} > n_i\right) \;\;\leq\;\; e^{-2\varepsilon^2 N}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\lim_{N \rightarrow \infty}\;\;
\hat{f}(x) &=
&f(x)\;\;\;\;\;\;\mbox{w.p. 1 and}\; \forall x\in {\rm dom}(f).
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\rho^{(n)} &=& \bar{\rho}^{n-1}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
 \frac{\bar{f}(x)
- f(x)}{f(x)} &<& \varepsilon 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
 \Delta ^{r}_{\pi }(x)&=&\left|\frac{\Delta _{\pi }(x)}{\pi (x)}\right| \nonumber\\\nonumber\\
 \Delta ^{r}_{\mu }(x|x')&=&\left|\frac{\Delta _{\mu }(x|x')}{\mu (x|x')}\right| \nonumber\\\nonumber\\
 \Delta ^{r}_{\nu }(O_{t}|x)&=&\left|\frac{\Delta _{\nu }(O_{t}|x)}{\nu (O_{t}|x)}\right| \nonumber\\\nonumber\\
 \Delta ^{r}_{f}(y)&=&\left|\frac{\Delta _{f}(y)}{f(y)}\right| 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta ^{r}_{\pi }&=&\max _{x}\Delta ^{r}_{\pi }(x) \nonumber\\\nonumber\\
\Delta ^{r}_{\mu }&=&\max _{x,x'}\Delta ^{r}_{\mu }(x|x') \nonumber\\\nonumber\\
\Delta ^{r}_{\nu }&=&\max _{x,O_{t}}\Delta ^{r}_{\nu }(O_{t}|x) \nonumber\\\nonumber\\
\Delta ^{r}_{f}&=&\max _{y}\Delta ^{r}_{f}(y) 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{f+g}(x) &=& \Delta _{f}(x)+\Delta _{g}(x) 
\label{Property1}\nonumber\\\nonumber\\
 \Delta _{\frac{g}{f}}(x) &=& \frac{-g(x)\Delta _{f}(x)+f(x)\Delta _{g}(x)}{f(x)^{2}} 
\label{Property2}\nonumber\\\nonumber\\ 
\Delta _{fg}(x) &=& f(x)\Delta _{g}(x) + g(x)\Delta _{f}(x) 
\label{Property3}\nonumber\\\nonumber\\ 
\Delta _{\int f}&=&\int \Delta _{f} 
\label{Property4} 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\alpha _{t}}&=&\Delta _{\int \alpha _{t-1}\mu \nu }(x)=\int _{x} \Delta _{\alpha _{t-1}\mu \nu }(x) \;dx
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray} 
&=&\int _{x} \mu (x)\nu (x)\Delta _{\alpha _{t-1}}(x)+\nu (x)\alpha _{t-1}(x)\Delta _{\mu }(x)+\mu (x)\alpha _{t-1}(x)\Delta _{\nu }(x) \;dx
\nonumber\nonumber\\\nonumber\\
&=&
\int _{x}\mu (x)\nu (x)\alpha _{t-1}(x)\left(\frac{\Delta _{\alpha _{t-1}}(x)}{\alpha _{t-1}(x)}+\frac{\Delta _{\nu }(x)}{\nu (x)}+\frac{\Delta _{\mu }(x)}{\mu (x)}\right) \;dx 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray} 
\Delta _{\alpha _{t}}&\leq& \alpha _{t}(x)\max _{x}\left(\left|\frac{\Delta _{\alpha _{t-1}}(x)}{\alpha _{t-1}(x)}+\frac{\Delta _{\nu }(x)}{\nu (x)}+\frac{\Delta _{\mu }(x)}{\mu (x)}\right|\right) 
\nonumber
\end{eqnarray} 
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray} 
\Delta _{\alpha _{t}}&\geq& -\alpha _{t}(x)\max _{x}\left(\left|\frac{\Delta _{\alpha _{t-1}}(x)}{\alpha _{t-1}(x)}+\frac{\Delta _{\nu }(x)}{\nu (x)}+\frac{\Delta _{\mu }(x)}{\mu (x)}\right|\right) 
\nonumber
\end{eqnarray} 
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray} 
\frac{\Delta _{\alpha _{t}}}{\alpha _{t}(x)} &\leq& \max _{x}\left(\left|\frac{\Delta _{\alpha _{t-1}}(x)}{\alpha _{t-1}(x)}\right|+\left|\frac{\Delta _{\nu }(x)}{\nu (x)}\right|+\left|\frac{\Delta _{\mu }(x)}{\mu (x)}\right|\right)
\nonumber\\\nonumber\\
\frac{\Delta _{\alpha _{t}}}{\alpha _{t}(x)} &\geq& - \max _{x}\left(\left|\frac{\Delta _{\alpha _{t-1}}(x)}{\alpha _{t-1}(x)}\right|+\left|\frac{\Delta _{\nu }(x)}{\nu (x)}\right|+\left|\frac{\Delta _{\mu }(x)}{\mu (x)}\right|\right)
\nonumber
\end{eqnarray} 
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\gamma }(x)&=&\Delta _{\frac{\alpha \beta }{\int \alpha \beta }}(x)
\nonumber\nonumber\\\nonumber\\
&=&\frac{\Delta _{\alpha \beta }(x)\int _{x} \alpha (x)\beta (x) dx 
- \alpha (x)\beta (x)\Delta _{\int \alpha \beta }(x)}{\left(\int _{x} \alpha (x)\beta (x)\right)^{2}}  \nonumber\nonumber\\\nonumber\\
\nonumber\nonumber\\\nonumber\\
&=&\frac{[\beta (x)\Delta _{\alpha }(x)+\alpha (x)\Delta _{\beta }(x)]\int _{x} \alpha (x)\beta (x) dx}{(\int _{x} \alpha (x)\beta (x) dx)^{2}}\nonumber\nonumber\\\nonumber\\
&&
\hskip 15mm
-\frac{\alpha (x)\beta (x)\int _{x} [\alpha (x)\Delta _{\beta }(x)+\beta (x)\Delta _{\alpha }(x)]dx }
{(\int _{x} \alpha (x)\beta (x) dx)^{2}}\nonumber\nonumber\\\nonumber\\
\nonumber\nonumber\\\nonumber\\
&\leq&
\frac{[\beta (x)\Delta _{\alpha }(x)+\alpha (x)\Delta _{\beta }(x)]\int _{x} \alpha (x)\beta (x)dx}{(\int _{x} \alpha (x)\beta (x)dx)^{2}}\nonumber\nonumber\\\nonumber\\
&&
\hskip 15mm
+\frac{\alpha (x)\beta (x)[\int _{x} \alpha (x)\beta (x) dx]\max _{x}\left(\left|\frac{\Delta _{\beta }(x)}{\beta (x)}\right|+\left|\frac{\Delta _{\alpha }(x)}{\alpha (x)}\right|\right)}{(\int _{x} \alpha (x)\beta (x)dx)^{2}}\nonumber\nonumber\\\nonumber\\
\nonumber\nonumber\\\nonumber\\
&\leq&
\alpha (x)\beta (x)\frac{\left[\frac{\Delta _{\alpha }(x)}{\alpha (x)}+\frac{\Delta _{\beta }(x)}{\beta (x)}\right]+\max _{x}\left(\left|\frac{\Delta _{\beta }(x)}{\beta (x)}\right|+\left|\frac{\Delta _{\alpha }(x)}{\alpha (x)}\right|\right)}{\int _{x}\alpha (x)\beta (x) dx}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\gamma }(x)
&\geq& - 
\alpha (x)\beta (x)\frac{\left[\frac{\Delta _{\alpha }(x)}{\alpha (x)}+\frac{\Delta _{\beta }(x)}{\beta (x)}\right]+\max _{x}\left(\left|\frac{\Delta _{\beta }(x)}{\beta (x)}\right|+\left|\frac{\Delta _{\alpha }(x)}{\alpha (x)}\right|\right)}{\int _{x}\alpha (x)\beta (x) dx}
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\gamma }^{r}&\leq& 2(\Delta ^{r}_{\alpha }+\Delta ^{r}_{\beta }) 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\gamma }^r &\leq& 2(\Delta ^{r}_{\pi }+(T-1)(\Delta ^{r}_{\mu }+\Delta ^{r}_{\nu }))
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\mu '}(x'|x)
&=&\Delta _{\frac{\sum \xi }{\sum \gamma }}(x'|x)\nonumber\nonumber\\\nonumber\\
&=&\left[-\left(\sum _{t=1}^{T - 1}\gamma _{t}(x)\right)\Delta _{\sum \xi }(x,x')+\left(\sum _{t=1}^{T - 1}\xi _{t}(x,x')\right)\Delta _{\sum \gamma }(x)\right]\frac{1}{\left(\sum _{t=1}^{T - 1}\gamma _{t}(x)\right)^{2}} \nonumber\nonumber\\\nonumber\\
&=& \left[-\left(\sum _{t=1}^{T - 1}\gamma _{t}(x)\right)\left(\sum _{t=1}^{T - 1}\Delta _{\xi _{t}}(x,x')\right)
\right.
\nonumber\nonumber\\\nonumber\\
&&\hskip 30mm \left.+\left(\sum _{t=1}^{T - 1}\xi _{t}(x,x')\right)\left(\sum _{t=1}^{T - 1}\Delta _{\gamma _{t}}(x)\right)\right]\frac{1}{\left(\sum _{t=1}^{T - 1}\gamma _{t}(x)\right)^{2}} \nonumber\nonumber\\\nonumber\\
&=&\left[\sum _{t=1}^{T - 1}\xi _{t}(x, x')\right]\;\left[-\frac{\sum _{t=1}^{T - 1}\Delta _{\xi _{t}}(x,x')}{\sum _{t=1}^{T - 1}\xi _{t}(x,x')}+\frac{\sum _{t=1}^{T - 1}\Delta _{\gamma _{t}}(x)}{\sum _{t=1}^{T - 1}\gamma _{t}(x)}\right]\;\frac{1}{\sum _{t=1}^{T - 1}\gamma _{t}(x)} \nonumber\nonumber\\\nonumber\\
&\leq& \left[\sum _{t=1}^{T - 1}\xi _{t}\right]\;\max _{t}\left(\frac{\Delta _{\xi _{t}}}{\xi _{t}}+\frac{\Delta _{\gamma _{t}}}{\gamma _{t}}\right)\frac{1}{\sum _{t=1}^{T - 1}\gamma _{t}} 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\mu '}(x'|x)
& \geq& - \left[\sum _{t=1}^{T - 1}\xi _{t}\right]\;\max _{t}\left(\frac{\Delta _{\xi _{t}}}{\xi _{t}}+\frac{\Delta _{\gamma _{t}}}{\gamma _{t}}\right)\frac{1}{\sum _{t=1}^{T - 1}\gamma _{t}} 
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\mu '}(x'|x)
& \leq& \mu(x'|x) \max _{t}\;\left(\frac{\Delta _{\xi _{t}}}{\xi _{t}}+\frac{\Delta _{\gamma _{t}}}{\gamma _{t}}\right)
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\Delta _{\mu '}(x'|x)
& \geq& - \mu(x'|x) \max _{t}\;\left(\frac{\Delta _{\xi _{t}}}{\xi _{t}}+\frac{\Delta _{\gamma _{t}}}{\gamma _{t}}\right)
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%

\begin{eqnarray}
\lim_{N\rightarrow \infty }\Delta _{\nu '}^{r}&=&0\nonumber\\\nonumber\\
\lim_{N\rightarrow \infty }\Delta _{\mu '}^{r}&=&0\nonumber\\\nonumber\\
\lim_{N\rightarrow \infty }\Delta _{\pi '}^{r}&=&0
\nonumber
\end{eqnarray}
\newpage

%-------------------------------------------------------------------------%


\end{document}

 

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