<p>In this paper, we consider a model associating random walks in a finite-dimensional Euclidean coordinate space of a classic system with random quantum walks, i.e., random transformations of the set of states of a quantum system arising from quantization of a classic system. It is known that the convolution semigroup of Gaussian measures on a coordinate space can be represented by a semigroup of self-adjoint contractions in the space of square-integrable functions described by the heat equation. We establish a representation of the convolution semigroup of Gaussian measures on the coordinate space by a quantum dynamic semigroup in the space of nuclear operators. We describe the quantum dynamic semigroup by solutions of the Cauchy problem for a degenerate diffusion equation. We prove the generalized convergence in distribution of a sequence of quantum random walks to an operator-valued random process with values in the Abelian algebra of shift operators by a vector with normal distribution.</p>