  We consider a one-dimensional continuous time random walk (CTRW) on a fixed
time interval $T$ where at each time step the walker waits a random time
$\tau$, before performing a jump drawn from a symmetric continuous probability
distribution function (PDF) $f(\eta)$, of L\'evy index $0 < \mu \leq 2$. Our
study includes the case where the waiting time PDF $\Psi(\tau)$ has a power law
tail, $\Psi(\tau) \propto \tau^{-1 - \gamma}$, with $0< \gamma < 1$, such that
the average time between two consecutive jumps is infinite. The random motion
is sub-diffusive if $\gamma < \mu/2$ (and super-diffusive if $\gamma > \mu/2$).
We investigate the joint PDF of the gap $g$ between the first two highest
positions of the CTRW and the time $t$ separating these two maxima. We show
that this PDF reaches a stationary limiting joint distribution $p(g,t)$ in the
limit of long CTRW, $T \to \infty$. Our exact analytical results show a very
rich behavior of this joint PDF in the $(\gamma, \mu)$ plane, which we study in
great detail. Our main results are verified by numerical simulations. This work
provides a non trivial extension to CTRWs of the recent study in the discrete
time setting by Majumdar et al. (J. Stat. Mech. P09013, 2014).
