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We give asymptotics for the cumulative distribution function (CDF) fordegrees of large dense random graphs sampled from a graphon. The proof is basedon precise asymptotics for binomial random variables. Replacing the indicatorfunction in the empirical CDF by a smoother function, we get general asymptoticresults for functionals of homomorphism densities for partially labeled graphswith smoother functions. This general setting allows to recover recent resultson asymptotics for homomorphism densities of sampled graphon.

Asymptotic for the cumulative distribution function of the degrees and homomorphism densities for random graphs sampled from a graphon