Perturbed self-intersection local time

We consider a symmetric random walk related to independent Rademacher random variables. Our aim is to study some modified versions of the so called self-intersection local time of this random walk. The modified versions of the self-intersection local time are obtained by introducing a time $t$ and a sequence of independent with the same distribution uniform on $(0,1)$ random variables $Y_i$'s, independent of the random walk. In this work, we study a distance between the standard self-intersection local time of the random walk and some modified versions (perturbed) of it. We also state a two-parameter strong approximation for the centered local time of the hybrids of empirical and partial sums processes by a process defined by a Wiener sheet combined with an independent Brownian motion.