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A property of Poisson process says this:

$N\left ( t \right )$ has independent increments:

if $t_{0}<\cdot \cdot \cdot <t_{n}$ then $N\left ( t_{1} \right )-N\left ( t_{0} \right ),\cdot \cdot \cdot ,N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ are independent.

I've been trying to understand this for many hours but unable to get around the physical intuition. My physical intuition has been really weak in getting my head around these ideas.

By some lemma: $N\left ( t \right )-N\left ( t_{n-1} \right )$ is independent of $N\left ( r \right ), r\leq t_{n-1}$.

This implies that $N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ is independent of $N\left ( t_{n-1} \right ),N\left ( t_{n-2} \right ),\cdot \cdot \cdot ,N\left ( t_{0} \right )$.

From here the author claims that "Hence, $N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ is independent of $N\left ( t_{n-1} \right )-N\left ( t_{n-2} \right ),\cdot \cdot \cdot ,N\left ( t_{1} \right )-N\left ( t_{0} \right )$" But I can't "see" this. It isn't at all "common sense" to me.

The only justification I may come up with is that

$N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ is independent of $N\left ( t_{n-1} \right ),N\left ( t_{n-2} \right )$ and so is independent of $N\left ( t_{n-1} \right ) - N\left ( t_{n-2} \right )$ under closure by subtraction, taking a leap.

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    $\begingroup$ There isn't any "lemma" involved: independent increments is part of the definition of a Poisson process. Perhaps the unnamed author has a weaker definition of independent increments? If so, what is it? $\endgroup$ – whuber Mar 15 '18 at 14:15
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The way I usually conceptualize this is to appreciate in a poison process, waiting times between events are distributed exponentially. Consequently, the hazard function is flat, which means that that the arrivals are independent of time. Since this is true, the counts of events within defined intervals are uncorrelated. This gives it the 'memoryless' or Markov property.

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$N(t)$ is the number of arrivals that occur after time $t =0$ and _up to and including time $t$, that is, the number of arrivals in the time interval $(0,t]$. It is a Poisson random variable with parameter $\lambda t$ where $\lambda$ is called the arrival rate. More generally, for $t_k < t_{k+1}$, $N(t_{k+1})-N(t_k)$ is the number of arrivals in $(t_k, t_{k+1}]$ and is a Poisson random variable with parameter $\lambda(t_{k+1}-t_k)$. As @whuber's comment says, the very notion of independent increments is that the increments (in the count of arrivals) that occur in disjoint intervals of time -- such as $(t_0,t_1],\quad (t_1, t_2],\quad \cdots \quad (t_{n-1},t_n]$ are mutually independent random variables, and this property is usually taken as part of the definition of a Poisson process. (The other property is the assumption that each increment is a Poisson random variable with parameter $\lambda \tau$ where $\tau$ is the length of the semi-open interval.

Turning to your question

"Hence, $N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ is independent of $N\left ( t_{n-1} \right )-N\left ( t_{n-2} \right ),\cdot \cdot \cdot ,N\left ( t_{1} \right )-N\left ( t_{0} \right )$" But I can't "see" this. It isn't at all "common sense" to me.

Note that on the previous line you happily accept that

This implies that $N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ is independent of $N\left ( t_{n-1} \right ),N\left ( t_{n-2} \right ),\cdot \cdot \cdot ,N\left ( t_{0} \right )$.

and consider that if $N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ is indeed independent of those $n$ random variables $N\left ( t_{n-1} \right ),N\left ( t_{n-2} \right ),\cdot \cdot \cdot ,N\left ( t_{0} \right )$, then it is also independent of any functions of those $n$ random variables. In particular, $N\left ( t_{n} \right )-N\left ( t_{n-1} \right )$ is independent of the $n$ increments $N\left ( t_{n-1} \right )-N\left ( t_{n-2} \right )$, $\cdot \cdot \cdot , N\left ( t_{1} \right )-N\left ( t_{0} \right )$.

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