Lyapunov Exponents Are not Rigid with Respect to Arithmetic Subsequences

We construct a random walk on the integers tending with positive velocity to infinity while any walk along an arbitrary arithmetic subsequence is recurrent. This can be interpreted as an example of a product of random matrices with positive Lyapunov exponent while any subproduct along an arithmetic subsequence has zero exponent. This shows that one can have exponential stretching along an orbit of a dynamical system while any time series along an arithmetic subsequence indicates subexponential growth.