Solution
= Solution
The running minima satisfy
$$
L_{n+1}=\min\{L_n,X_{n+1}\}.
$$
Since $X_{n+1}$ is independent of the past, the conditional law of $L_{n+1}$ depends only on $L_n$. Hence $(L_n)$ is a Markov chain. This is the <running minimum of an independent sequence is Markov> property.