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 $ψ$-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 $\{0, 1\}^\mathbb{N}$.