We have investigated the geometry properties of the random fractals derived from two independent Lévy processes.The first main result is the following: Let X1 and X2 be two independent general subordinators with lower (upper) indices σ1 and σ2 (β1 and β2) respectively.Denote the occupation measure of Xi by μi (i =1, 2).Set μ =μ1 ×μ2 and denote B(x, r) as the ball centered at x with radiusr.