ON THE HITTING PROBABILITIES OF LIMSUP RANDOM FRACTALS

Let $A$ be a limsup random fractal with indices $\gamma_1, ~\gamma_2 ~$and $\delta$ on $[0,1]^d$. We determine the hitting probability $\mathbb{P}(A\cap G)$ for any analytic set $G$ condition $(\star)$$\colon$ $\dim_{\rm H}(G)>\gamma_2+\delta$, where H}$ denotes Hausdorff dimension. This extends correspondence of Khoshnevisan, Peres and Xiao [10] by relaxing that $P_n$ choosing each dyadic hyper-cube is homogeneous $\lim\limits_{n\to\infty}\frac{\log_2P_n}{n}$ exists. also present some counterexamples to show dimension in $(\star)$ can not replaced packing