If 2 random variables X, and Y have the same distribution, does that mean that for another random variables Z, X + Z and Y + Z also have the same distribution? From looking at the convolution formula, the answer should be yes, because the convolution of random variables depends only on the distribution of the 2 variables being added. However, if we consider the Poisson process N(t), then the stationary increment property says that N(5) - N(4) has the same distribution as N(1). Now consider the random variable (N(4) - N(5)) - N(1). If the above claim is true then this has the same distribution as N(1) - N(1), which has variance 0, i.e. Var((N(4) - N(5)) - N(1)) = 0. This would seem to imply that the count of random Poisson events between time 4 and 5 must always be exactly the same as between time 0 and 1 and there is actually no randomness at all, which can't possibly be true.