A further (and final) definition of the Poisson process runs as follows: A nondecreasing stochastic process {X(t), t ≥ 0} is a Poisson process iff
(a) it is nonnegative, integer-valued, and X(0) = 0;
(b) it has independent, stationary increments;
(c) it increases by jumps of unit magnitude only.
Show that a process satisfying these conditions is a Poisson process.
Remark. Note that if {X(t), t ≥ 0} is a Poisson process, then conditions (a)-(c) are obviously satisfied. We thus have a fourth, equivalent, definition of a Poisson process.