I'm trying to understand the significance of the memoryless properties for Markov processes and for the exponential distribution. Both this answer and this answer have been very helpful. However, I am still not sure if this is correct:
For a stochastic process $(X(t))_{t\geq 0}$ with discrete state space, do exponentially distributed holding times $T_i \sim \exp(\lambda)$ imply that the process is a Markov process?
I would argue that the answer is yes as at any given time point $T$ the probability for no jump to occur until $T+s$ is
\begin{align} P(X_{T+s}=x_T|X_t,t\leq T) &= P(T_{i}>T+s|X_T=x_T) \\ &= P(T_{i}>T+s|T_{i}>T) \\ &= P(T_{i}>s) \\ &= e^{- \lambda s} \end{align}
(where $T_{i}$ is the holding time of the current state). So this probability only depends on the increment, which would mean not only is the process Markovian, but also homogeneous. Is that correct?