Asymptotic Statistics Of Random Polynomials Over Finite Fields

In this paper,we apply the ideas of randomness to polynomial rings over finite fields,and consider the asymptotic statistics of random polynomials.It includes two parts.To be explicit,in the first part,we establish,by the dependency graphs and Stein's methods,the central limit theorems and moderate deviations for the empirical density of the coprime random polynomials,which extend the partial results of Fernandez&Fernandez(Int.J.Number Theory,2015)and Mehrdad&Zhu(Q.J.Math.,2016)to the polynomial rings.In the second part,we discuss the completely additive functions on polynomial rings,and give the necessary and sufficient conditions for its convergence to the infinitely divisible distributions,which extend some results of Billingsley(Ann.Probab.,1974)to the polynomial rings.