Abstract: In this paper the authors discuss the Holder laws for self-intersection local time increments of N-parameter d-dimension generalized α-stable
process, prove the existence of multiple points, and obtan the lower bounds about Hausdorff dimension and Hausdorff measure of multiple time. The obtained results contain and extend the existing results of
generalized Brownian sheet and Stable sheet.

Key words: Self-intersection local time, H\"older law, α-stable process, Multiple point