In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $μ$ on $M$ where the transition kernel $\mathcal P$ admits an eigenfunction $0\leq η\in L^1(M,μ)$. We find conditions on the transition densities of $\mathcal P$ with respect to $μ$ which ensure that $η(x) μ(\mathrm d x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $μ$-almost surely. We apply this result to the random logistic map $X_{n+1} = ω_n X_n (1-X_n)$ absorbed at $\mathbb R \setminus [0,1],$ where $ω_n$ is an i.i.d sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$