  In recent years, generative adversarial networks (GANs) have demonstrated
impressive experimental results while there are only a few works that foster
statistical learning theory for GANs. In this work, we propose an infinite
dimensional theoretical framework for generative adversarial learning. We
assume that the probability density functions of the underlying measure are
uniformly bounded, $k$-times $\alpha$-H\"{o}lder differentiable
($C^{k,\alpha}$) and uniformly bounded away from zero. Under these assumptions,
we show that the Rosenblatt transformation induces an optimal generator, which
is realizable in the hypothesis space of $C^{k,\alpha}$-generators. With a
consistent definition of the hypothesis space of discriminators, we further
show that the Jensen-Shannon divergence between the distribution induced by the
generator from the adversarial learning procedure and the data generating
distribution converges to zero. Under certain regularity assumptions on the
density of the data generating process, we also provide rates of convergence
based on chaining and concentration.
