I'm trying to find the expectation of a function conditional on being on a particular subset of $\mathbb{R}^2$, under the Laplace distribution. The key step, however, is that the subset is moving. So the limit I want to calculate is \begin{equation} \lim_{t\rightarrow \infty }E\left[ D\left( x_{1}/t,x_{2}/t\right) \mid \min \left( x_{1},x_{2}\right) =-t\right] \end{equation} where $x_{1}$ and $x_{2}$ are independently Laplace distributed with scale $% \lambda $. Make whatever assumptions you need about $D$. For example, it might be that $D\left( x,y\right) =1$ for $\left\vert x\right\vert <b$ and $% \left\vert y\right\vert <b$ and $0$ otherwise, for some small $b$.
The expectation can be written as an integral, \begin{equation} \lim_{t\rightarrow \infty }\frac{\int_{\min \left( x_{1},x_{2}\right) =-t}D\left( x_{1}/t,x_{2}/t\right) \exp \left( -\lambda \left( \left\vert x_{1}\right\vert +\left\vert x_{2}\right\vert \right) \right) dV}{\int_{\min \left( x_{1},x_{2}\right) =-t}\exp \left( -\lambda \left( \left\vert x_{1}\right\vert +\left\vert x_{2}\right\vert \right) \right) } \end{equation} A change of variables with $y_{j}=x_{j}/t$ yields \begin{equation} \lim_{t\rightarrow \infty }\frac{\int_{\min \left( y_{1},y_{2}\right) =-1}D\left( y_{1},y_{2}\right) \exp \left( -\lambda t\left( \left\vert y_{1}\right\vert +\left\vert y_{2}\right\vert \right) \right) dV}{\int_{\min \left( y_{1},y_{2}\right) =-1}\exp \left( -\lambda t\left( \left\vert y_{1}\right\vert +\left\vert y_{2}\right\vert \right) \right) } \end{equation} I don't know what to do with that. My intuition is that it should converge to the average of $D$ at two points, $(0,-1)$ and $(-1,0)$.