A sequence \(\{P_ i\}\) of real Laurent polynomials with nonnegative coefficients is said to be strongly positive if, for any real Laurent polynomial \(f\) with \(f|_{\mathbb{R}^ +}>0\) and for every integer \(k\), there exists an integer \(n\) such that \(P_ k\cdot P_{k+1} \cdots P_{k+n}f\) has nonnegative coefficients. This can be rephrased in terms of a random walk problem for random variables with values in \(\mathbb{Z}\) (just associate with a polynomial \(P(x)=\sum c_ i x^ i\), \(i\in\mathbb{Z}\), \(a_ i\in\mathbb{R}\), \(a_ i=0\) for almost all \(i\), a random variable \(X\) with \(\text{Prob}(X=j)=c_ j/P(1)\)). The authors prove necessary and sufficient conditions for strong positivity. Moreover, they discuss the pure harmonic functions \(h:\mathbb{Z}\times\mathbb{N}\to\mathbb{R}^ +\) on the random walk (with values in \([0,\infty]\)), and show that, under the assumption of strong positivity, all such functions are point evaluations of the form \(h_ r(m,i)=r^ m/P_ 1\cdot P_ 2\cdots P_ i(r)\), \(r>0\), or limit points of these. Further conditions for the purity of individual point evaluations are discussed.