Generating infinite random graphs

We define a growing model of random graphs. Given a sequence of nonnegative integers $\{d_n\}_{n=0}^\infty$ with the property that $d_i\leq i$, we construct a random graph on countably infinitely many vertices $v_0,v_1\ldots$ by the following process: vertex $v_i$ is connected to a subset of $\{v_0,\ldots,v_{i-1}\}$ of cardinality $d_i$ chosen uniformly at random. We study the resulting probability space. In particular, we give a new characterization of random graph and we also give probabilistic methods for constructing infinite random trees.