Let K be a finite field of characteristic p . Let K ( ( x ) ) be the field of formal Laurent series f ( x ) in x with coefficients in K . That is, f ( x ) = ∑ n = n 0 ∞ f n x n with n 0 ∈ 𝐙 and f n ∈ K ( n = n 0 , n 0 + 1 , ⋯ ) . We discuss the distribution of ( { f m } ) m = 0 , 1 , 2 , ⋯ for f ∈ K ( ( x ) ) , where { f } : = ∑ n = 0 ∞ f n x n ∈ K [ [ x ] ] denotes the nonnegative part of f ∈ K ( ( x ) ) . This is a little different from the real number case where the fractional part that excludes constant term (digit of order 0) is considered. We give an alternative proof of a result by De Mathan obtaining the generic distribution for f with f n ≠ 0 for some n < 0 . This distribution is not the uniform measure on K [ [ x ] ] , but is equivalent to it. We have a different situation for f ∈ K [ [ x ] ] , where if f 0 ≠ 0 and f ≠ f 0 , then the distribution for f is continuous but has a small support. We prove in this case, that the distribution for f - 1 is identical with the distribution for f 0 - 2 f . Christol, Kamae, Mendès France and Rauzy proved that the algebraicity of f ( x ) ∈ K ( ( x ) ) over K ( x ) is equivalent to the p -automaticity of the sequence ( f n ) . This result was generalized to the multidimensional case by Salon. Hence, if the Laurent series f ( x ) ∈ K ( ( x ) ) is algebraic over K ( x ) , then F ( x , y ) : = ∑ m = 0 ∞ f ( x ) m y m is 2 -dimensionally p -automatic, since it is algebraic over the field K ( x , y ) . We construct a finite automaton recognizing the sequence of coefficients of this double series F ( x , y ) to discuss the distribution of ( { f m } ) m ≥ 0 . Thus, we generalize results by Houndonougbo and Deshouillers, and strengthen results by Allouche and Deshouillers.