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

On the quasi-ergodicity of absorbing Markov chains with unbounded transition densities, including random logistic maps with escape