# Independent increments vs independent events

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

• Independent increments means that the number of events in any two disjoint intervals are independent. What do you mean by independent events of the Poisson process? – QQQ Oct 1 '18 at 18:35
• Sorry for the lack of clarification.When I say two events are independent, I mean that the occurrence of one event doesn't affect the probability of another. So if A and B are independent events, P (A)=P (A | B) – Ecoboi Oct 2 '18 at 8:20
• Yes, but what events specifically are you asking about in this context? The independent increments means that ${N(t_2)-N(t_1)}$ and ${N(t_4)-N(t_3)}$ are independent for $t_1<t_2<t_3<t_4$. What events are you asking if this implies independence for? – QQQ Oct 2 '18 at 19:56
• A counting process {N (t)} is one where N (t) represents the number of events that had occurred up to time t. So for instance, if the counting process was a measure of the number of insurance claims which had occurred, an event would be a claim occurring – Ecoboi Oct 3 '18 at 0:10
• "A claim occurring" is not a well defined event (in the probabilistic meaning). For example, "a claim occurring before time $t$" is an event, and it is independent of events such as "10 claims occurred during $[s,u]$" where $s>t$, but not independent of the event if $s<t$. Are you perhaps thinking of the independence of the time between claims? – QQQ Oct 3 '18 at 7:18