Viktor Schulmann, TU Dortmund
Life span estimation for randomly moving particles based on their places of death
Consider the following problem from physics: A radiation source is placed at the center of a screen. At certain time intervals the source releases particles. These move around the screen following a path of some known random process $(Y_t)_{t\geq 0}$ without interacting with each other and without us being able to observe their movement until they die after some random time $T$. During its death a particle leaves a mark such that we can measure the distance $X=Y_T_2$ it traveled from the source during its lifetime. Based on these observed distances we wish to infer the life span $T$ of a particle or, in particular, the density $f_T$ of $T$.
We assume $(Y_t)_{t\geq 0}$ from our physics experiment to be a multidimensional L\'{e}vy processes with spherical symmetry. Norms of such processes exhibit structural similarities to onedimensional L{\'e}vy processes. For that case an estimator was given by Belomestny and Schoenmakers (2016) using the Mellin and Laplace transforms. Applying their techniques we construct a nonparametrical estimator for $f_T$, calculate its convergence rate and show its optimality in the minimax sense.
