Question

1. LetXbe a uniform random variable on (0,1), and consider a counting process where eventsoccur at timesX+i, fori= 0,1,2,....(a) Does this counting process have independent increments?(b) Does this counting process have stationary increments?

Answer 1

(a) No.

(b) Yes.

Explanation:

(a)

Let N(t) denote the number of points on the interval [0,t].

Consider the random variables N(1/4) and N(1/2)-N(1/4).

P(N(1/4) = 1) = P(0<X≤1/4) = 1/4,

P(N(1/2)-N(1/4) = 1) = 1/4.

However,

P(N(1/4) = 1, N(1/2)-N(1/4) = 1) = 0,

as the process cannot have two points on the interval [0,1/2].

Since 0 = P(N(1/4) = 1),

P(N(1/2)-N(1/4) = 1) = 6 ≠ P(N(1/4) = 1) * P(N(1/2)-N(1/4) = 1) = 1/16,

the counting process does not have independent increments

(b)

N(s+t)-N(s) and N(t) have the same distribution.

First consider N(t). Since the distance between points in the counting process is exactly 1,

N(t) must have either [t] or [t]+1 points, where [x] is the greatest integer less than or equal to x.

Since N(t)=[t]+1 only if there is a point on the interval ([t],t)

by this the counting process have stationary increments

Hope this helps!