If Xn is a simple
random walk on the nonnegative integers with transition probabilities
Pij(k)= Pr{Xn+k= j∣Xn= i}, then Pij(k) has an integral representation in terms
of a family of orthogonal polynomials and the associated probability distribution
function F(x) for these polynomials. The relationship between the distribution F,
the family of polynomials and the random walk Xn is studied. Necessary
and sufficient conditions for the support of F to be contained in [0,1] are
given.