Erdös-Rényi law for the local time of the hybrid process

Our aim in this paper is to study the Erdös-Rényi law for the local time of the hybrid of empirical and partial sums process. The corresponding local time can be see as a modified version of the local time of the symmetric random walk by introducing a time $t$ and a sequence of independent with the same distribution random variables $X_i$'s, independent of the random walk.