I am considering a counting stochastic process which has independent increments (just like a Poisson process).
We know that a Poisson process has independent increments. Furthermore, one of the assumptions is that the events being counted are independent of each other (since the number of events occurring between any two times is assumed to follow a Poisson distribution, which implies independence of events).
I was wondering whether the property of independent increments for a stochastic counting process (in general) also implies that the events being counted are independent of each other? I understand that the converse is true, but I am interested in the forward direction.
Thank you