I understand that inter-arrival times of a Poisson process are exponentially distributed and therefore the inter-arrival times are memoryless.
However, how about the waiting times of Poisson process i.e wait time till $k$th arrival where $k \geq 2$. This is an Erlang distribution and shouldn't be memoryless -- or is it?
If it is, can someone help how to show it?
In general, why is Poisson process memoryless? I understand that interarrival times or time to next arrival are but doesn't look like time till kth arrival is also memoryless .. or is it?