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After providing an overview of $\theta$-expansions introduced by Chakrabortyand Rao, we focus on the Gauss-Kuzmin problem for this new transformation.Actually, we complete our study on these expansions by proving atwo-dimensional Gauss-Kuzmin theorem. More exactly, we obtain such a theoremrelated to the natural extension of the associated measure-dynamical system.Finally, we derive explicit lower and upper bounds of the error term whichprovide interesting numerical calculations for the convergence rate involved.

On convergence rate in the Gauss-Kuzmin problem for $\theta$-expansions