In reading Ross' Introduction to Probability Models, there's a part on mixed Poisson processes which states
Suppose that $L$ is continuous with density function $g$. Because
$P\{N(t+s) - N(s) = n \} = \int_0^{\infty} P\{N(t+s) - N(s) = n | L = \lambda \}\ g(\lambda)\ d\lambda = \int_0^{\infty} e^{-\lambda t}\frac{(\lambda t)^n}{n!}\ g(\lambda)\ d\lambda$
we see that a conditional Poisson process has stationary increments. However, because knowing how many events occur in an interval gives information about the possible value of $L$, which affects the distribution of the number of events in any other interval, it follows that a conditional Poisson process does not generally have independent increments. (...)
I don't understand why "it follows that a conditional Poisson process does not generally have independent increments". Can someone give a concrete example? And in what instances does it have independent increments?