Question

Let \(X\) be a uniform random variable on \((0,1)\), and consider a counting process where events occur at times \(X+i\), for \(i=0,1,2, \ldots\) (a) Does this counting process have independent increments? (b) Does this counting process have stationary increments?

Short answer

(a) This counting process does not have independent increments because the probabilities of events occurring in non-overlapping intervals depend on \(i\) and \(j\) and are not independent. (b) This counting process does not have stationary increments because the distribution of increments depends not only on the length of the time interval, but also on the specific starting and ending points of the interval.

Step-by-step solution

(a) Independent Increments#

We will verify if the counting process has independent increments by checking if the number of events occurring in non-overlapping intervals are independent of each other. Consider two non-overlapping intervals \([t_1, t_2)\) and \([t_3, t_4)\) with \(t_2 < t_3\). Let's find the probability that an event occurs in each interval. An event occurs at time \(X+i\) in the interval \([t_1,t_2)\) if \(t_1<X+i<t_2\). Since \(X\) is a uniform random variable on \((0,1)\), the probability of an event occurring in the interval \([t_1,t_2)\) is \[ P(t_1 < X+i < t_2) = P(t_1 -i < X < t_2-i). \] Similarly, for the interval \([t_3, t_4)\), the probability of an event occurring is \[ P(t_3 < X+j < t_4) = P(t_3 -j < X < t_4-j). \] These probabilities depend on \(i\) and \(j\) and are not independent of each other. Thus, this counting process does not have independent increments.

(b) Stationary Increments#

We will verify if the counting process has stationary increments by checking if the distribution of the increments depends only on the length of the time interval and not on the specific starting and ending points of the interval. Consider a fixed time interval of length \(\tau\), say \([0, \tau)\), and an event occurring at time \(X+i\) in this interval. For a fixed \(i\), the probability that an event occurs in the interval \([0, \tau)\) is \[ P(0 < X+i < \tau) = P(-i < X < \tau - i). \] Now consider the same time interval of length \(\tau\) but starting at a different time, say \([t, t+\tau)\). The probability of an event occurring at time \(X + i\) in this interval is \[ P(t < X+i < t + \tau) = P(t-i<X<t+\tau-i). \] By comparing these probabilities, we can see that the distribution of the increments depends not only on the length of the time interval \(\tau\) but also on the specific starting time \(t\). Thus, this counting process does not have stationary increments.