\begin{equation}
\begin{aligned}
E\left ( \varepsilon _{n}\right )
&=E\left ( v_{n}t_{n} \right )+E\left ( X_{n}^{\left ( v_{n} \right )} \right )\\
&=E\left ( v_{n}t_{n} \right )+E\left [ E\left ( X_{n}^{\left ( v_{n} \right )}\mid v_{n} \right ) \right ]\\
&=\frac{R}{1-R}\cdot \frac{t_{1}}{a^{n-1}}+\frac{1}{1-R} \int_{0}^{t_{n}}tdF\left ( a^{n-1}t \right )\\
&=\frac{1}{\left ( 1-R \right )a^{n-1}}\int_{0}^{t_{1}}\left ( 1-F\left ( t \right ) \right )dt
\end{aligned}
\end{equation}


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