In this paper we consider a random star $d$-process which begins with $n$ isolated vertices, and in each step chooses randomly a vertex of current minimum degree $\delta$, and connects it with $d - \delta$ random vertices of degree less than $d$. We show that, for $d \geqslant 3$, the resulting final graph is connected with probability $1 - o(1)$, and moreover that, for suficiently large $d$, it is $d$-connected with probability $1 - o(1)$.