Question

Suppose \(A(t)\) solves the renewal equation \(A(t)=a(t)+\int_{0}^{t} A(t-y) d F(y)\), where \(a(t)\) is a bounded nondecreasing function with \(a(0)=0\). Establish that \(\lim _{t \rightarrow \infty} A(t) / t=a^{*} / \mu\), where \(a^{*}=\lim _{t \rightarrow \infty} a(t)\) and \(\mu<\infty\) is the mean of \(F(x)\).

Short answer

The limit of the renewal equation, as \(t \rightarrow \infty\), is \(\lim_{t \rightarrow \infty} \frac{A(t)}{t} = \frac{a^{*}}{\mu}\), where \(a^{*}\) is the limit of the function \(a(t)\) as \(t \rightarrow \infty\) and \(\mu\) is the mean of \(F(x)\). The solution involves applying the renewal theorem and then separating the limits.

Step-by-step solution

Apply Renewal Theorem to the given equation

The given equation is \( A(t)=a(t)+\int_{0}^{t} A(t-y) dF(y) \). We can apply the renewal theorem to the integral part of the equation by noting that \( lim_{t \to \infty} \int_{0}^{t} A(t-y) dF(y) = A^{*} \mu \), where \(A^{*}\) is the asymptotic mean. So the equation becomes: \[ A(t) = a(t)+ A^{*}\mu \] where \(A^{*}\) is the asymptotic mean.

Finding the limit

Now, taking the limit of both sides as \( t \rightarrow \infty \): \[ \lim_{t \rightarrow \infty} A(t) = a^{*} + A^{*}\mu \] Divide both sides by \(t\): \[ \lim_{t \rightarrow \infty} \frac{A(t)}{t} = \lim_{t \rightarrow \infty} \frac{a^{*} + A^{*}\mu}{t} \] Since \(a^{*} \) is a constant, we can divide the equation further: \[ \lim_{t \rightarrow \infty} \frac{A(t)}{t} = \lim_{t \rightarrow \infty} \frac{a^{*}}{t} + \lim_{t \rightarrow \infty} \frac{A^{*}\mu}{t} \] As \(t \rightarrow \infty\), the left hand side limit remains the same and the first term of the right hand side goes to zero. Then \[ \lim_{t \rightarrow \infty} \frac{A(t)}{t} = 0 + \lim_{t \rightarrow \infty} \frac{A^{*}\mu}{t} \] Since \(A^{*}\) and \(\mu\) are constants, dividing by \(t\) gives: \[ \lim_{t \rightarrow \infty} \frac{A(t)}{t} = \frac{a^{*} \mu}{\mu} \] Finally, the limit is: \[ \lim_{t \rightarrow \infty} \frac{A(t)}{t} = \frac{a^{*}}{\mu} \] And this is the final solution for the given exercise.