Random walks on the non-negative integers with transition probability \(p_{0,1}=1\), \(p_{k,k-1}=g_ k\), \(p_{k,k+1}=1-g_ k\), \(0<g_ k<1\), \(k=1,2,...\), are studied by using the continued fraction expansions of generating functions \(F(z)=\sum^{\infty}_{n=0}f_{00}^{(2n)}z^ n\) and \(G(z)=\sum^{\infty}_{n=0}p_{00}^{(2n)}z^ n\). The stress is put on the case when \((g_ k)\) is periodic with period p. The main result is the following local limit theorem: \[ p_{jk}^{(m+k-j)}\sim A_{jk}\quad if\quad r_ p<1,\quad =B_{jk}n^{-1/2}\quad if\quad r_ p=1,\quad =C_{jk}R^{-2n}n^{-3/2}\quad if\quad r_ p>1, \] where \(r_ p=(1-g_ 1)(1-g_ 2)...(1-g_ p)(g_ 1...g_ p)^{-1}\), \(R^ 2=s\) is the radius of convergence of F(z) and A’s, B’s, C’s are constants. It is also shown how to calculate these constants explicitly.