Abstract: In this paper, a class of generalized random recursive construction with finite memory in Euclidean $d$-space is researched. For each $\beta\geq 0$, a function $\Psi(\beta)$ assiociated with the construction is introduced and a random measure $\mu_{\omega}$ is constructed. That the Hausdorff dimension of the random limit set $K(\omega)$ generated by the above construction is equal to $\alpha:=\inf\{\beta:\Psi(\beta)\leq1\}$ is proved.

Key words: Hausdorff dimension, Random construction, Supermartingale, Extension of measure, Random measure, Local dimension