Question

Show that the age \(\left\\{\delta_{t} ; t \geq 0\right\\}\) in a renewal process, considered as a stochastic process, is a Markov process, and derive its transition distribution function $$ F(y ; t, x)=\operatorname{Pr}\left\\{\delta_{s+t} \leq y \mid \delta_{s}=x\right\\} $$

Short answer

To prove that the age \( \delta_t \) in a renewal process is a Markov process, we showed that the probability of future age depends only on the current age and not on past history and that the Markov property is satisfied as the residual life depends only on the current age. Then, we derived the transition distribution function and found that $$ F(y;t,x) = \frac{G(t + x - y) - G(x)}{1 - G(x)} , $$ where \(G(t)\) denotes the interarrival time distribution function.

Step-by-step solution

Understanding the renewal process

A renewal process is a point process where events occur at random moments in time, with the time elapsed between two events, called interarrival times, following a probability distribution. The age \( \delta_t \) at time \(t\) is the time elapsed since the last event before time \(t\).

Proving the Markov property

To show that the age in a renewal process is a Markov process, we need to prove that the probability of the age \( \delta_{t+s} \) being less than or equal to a given value \(y\) depends only on the current age \( \delta_s \) and not on any previous age information. Mathematically, this is shown as: $$ Pr(\delta_{s+t} \leq y | \delta_s = x, \delta_{s_1} = x_1, \cdots, \delta_{s_n} = x_n) = Pr(\delta_{s+t} \leq y | \delta_s = x) $$ where \(s_1, \cdots, s_n\) are any previous time points and \(y\) is a given value. To prove this, let \(R_t\) denote the residual life at time \(t\), that is the amount of time until the next event occurs. Since the interarrival times in a renewal process are memoryless, the residual life only depends on the current age and not on past history. Then, given that the current age is \(x\), the probability that the age at time \(s + t\) is less than or equal to \(y\) is the same as the probability that the residual life at time \(s\) is greater than \(t - y\). Mathematically, we have: $$ Pr(\delta_{s+t} \leq y | \delta_s = x) = Pr(R_s > t - y | \delta_s = x) $$ Since the residual life depends only on the current age, the Markov property is satisfied.

Deriving the transition distribution function

Finally, we need to find the transition distribution function \(F(y;t,x)\). We have already shown that: $$ F(y;t,x) = Pr(\delta_{s+t} \leq y | \delta_s = x) = Pr(R_s > t - y | \delta_s = x) $$ Now, let's denote the interarrival time distribution function by \(G(t)\). We know that: $$ G(t) = Pr(\delta_{s} \leq t) = 1 - Pr(R_s > t) $$ So, $$ Pr(R_s > t) = 1 - G(t) $$ Given that the current age is \(x\), the probability that the residual life at time \(s\) is greater than \(t - y\) is equal to the probability that a random interarrival time is greater than \((t - y) + x\), conditional on the initial interarrival time being greater than \(x\). Thus, we have: $$ F(y;t,x) = Pr(\delta_{s+t} \leq y | \delta_s = x) = \frac{Pr(R > (t - y) + x | R > x)}{Pr(R > x)} $$ Now, using the fact that: $$ Pr(R > t) = 1 - G(t) $$ We can write the transition distribution function as: $$ F(y;t,x) = \frac{1 - G((t - y) + x) - (1 - G(x))}{1 - G(x)} $$ Simplifying, we get the final expression for the transition distribution function as: $$ F(y;t,x) = \frac{G(t + x - y) - G(x)}{1 - G(x)} $$

Key concepts

Understanding Renewal Processes

Imagine we have a series of events that occur one after another at unpredictable times. Think of a light bulb that burns out and gets replaced, only to eventually burn out again. Each time we replace the bulb, it starts a new 'cycle'. A renewal process is a mathematical model of these kinds of events.

In this model, the time between the 'burn-outs' is called an interarrival time. Each of these times is a random variable that can follow any probability distribution, but they are often assumed to be independent and identically distributed (i.i.d). The 'age' at any time refers to the time elapsed since the last event occurred. One of the significant aspects of renewal processes is its pattern that somewhat 'resets' the situation each time a new cycle begins, which leads to interesting mathematical properties.

Students often find this concept easier to understand by visualizing or simulating a renewal process, seeing how after each renewal (like changing the light bulb), the process starts anew without being affected by when the previous bulbs were changed.

The Markov Process Nature of Renewal Processes

Now let's delve into the concept of a Markov process. This is a type of stochastic process that has a specific property: its future state depends only on its current state and not on its past states.

When we relate this to our 'light bulb' scenario in the renewal process, it means that if we want to predict when the next bulb will burn out, all we need to know is how long the current bulb has been on. We don't need to worry about how long previous bulbs lasted. This idea is called the memoryless property, which is a hallmark of the Markov process.

To show that the 'age' in a renewal process is a Markov process, we use the concept of 'residual life', which is the time remaining until the next 'burn-out'. If the distribution of interarrival times is memoryless, the residual life at present depends only on the current 'age' and not on the entire history, thereby satisfying the Markov property.

Understanding Markov processes can become clearer through exercises that involve drawing state diagrams or through the analysis of simple scenarios, such as games of chance, which adhere to Markovian principles.

Deriving the Transition Distribution Function

The transition distribution function is paramount in characterizing the behavior of a Markov process. For our renewal process, it represents the probability that the 'age' of the process at a future time (let's call this t+ s) is less than or equal to a certain value (y), given its current 'age' (x).

To derive it, we start by connecting the probabilities of the residual life with the interarrival time distribution, which is typically denoted as G(t). Since the future state depends only on the present in a Markov process, the equation to calculate the transition distribution function reduces to a matter of subtracting probabilities associated with the interarrival times, making it a ratio that normalizes the result by the probability of the remaining time being greater than the current 'age'.

When students work through the derivation step by step, seeing how the formula simplifies through algebraic manipulation, they often gain a deeper understanding of the process. It's not just an abstract formula; it's a way of predicting future events in a renewal process based on the current state, which beautifully illustrates the power of the Markov property.