  The results of this paper are 3-folded. Firstly, for any stationary
determinantal process on the integer lattice, induced by strictly positive and
strictly contractive involution kernel, we obtain the necessary and sufficient
condition for the $\psi$-mixing property. Secondly, we obtain the existence of
the $L^q$-dimensions of the stationary determinantal measure on symbolic space
$\{0, 1\}^\mathbb{N}$ under appropriate conditions. Thirdly, the previous two
results together imply the precise increasing rate of the longest common
substring of a typical pair of points in $\{0, 1\}^\mathbb{N}$.
