In this paper, we consider absorbing Markov chains Xn admitting a quasi-stationary measure μ on M where the transition kernel P admits an eigenfunction 0≤η∈L1(M,μ). We find conditions on the transition densities of P with respect to μ which ensure that η(x)μ(dx) is a quasi-ergodic measure for Xn and that the Yaglom limit converges to the quasi-stationary measure μ-almost surely. We apply this result to the random logistic map Xn+1=ωnXn(1−Xn) absorbed at R∖[0,1], where ωn is an independent and identically distributed sequence of random variables uniformly distributed in [a,b], for 1≤a<4 and b>4.