In this paper, the distribution function
and the generating function of
φ(u+1) are set up. We assume that
u∈N∪{0},
κ∈N, the random walk
{∑ni=1Xi,n∈N} involves
N∈N periodically occurring distributions, and the integer-valued and nonnegative random variables
X1,X2,… are independent. This research generalizes two recent works where
{κ=1,N∈N} and
{κ∈N,N=1} were considered respectively. The provided sequence of sums
{∑ni=1(Xi−κ),n∈N} generates the so-called multi-seasonal discrete-time risk model with arbitrary natural premium and its known distribution enables to compute the ultimate time ruin probability
1−φ(u) or survival probability
φ(u). The obtained theoretical statements are verified in several computational examples where the values of the survival probability
φ(u) and its generating function are provided when
{κ=2,N=2},
{κ=3,N=2},
{κ=5,N=10} and
Xi adopts the Poisson and some other distributions. The conjecture on the nonsingularity of certain matrices is posed.