I am analyzing a renewal process $N(t)$ whose inter arrival time $t$ conforms to the Pareto distribution, the PDF $f(t)$ of the Pareto distribution is as follows:
$$f(t) = \left\{ \begin{array}{lr} \frac{{\alpha {k^\alpha }}}{{{t^{\alpha + 1}}}},&t > k\\ 0,&t < k \end{array} \right.$$ where $\alpha$ is the shape parameter, $k$ is the scale parameter ($\alpha>0,k>0$).
The CDF $F(t)$ is:
$$F(t) = \left\{ \begin{array}{lr} 1 - {\left( {\frac{k}{t}} \right)^\alpha },&t > k\\ 0,&t < k \end{array} \right.$$
Let $m(t)=E[N(t)]$, $m(t)$ is referred to as the renewal function, and let $\tilde{m}(s)$ denote the Laplace-Stieltjes Transform (LST) of $m(t)$, so
$$\tilde{m}(s)=\int_0^\infty e^{-st}\mathrm dm(t)$$
It's well known that $m(s)$ satisfies the following equation (the LST of the renewal equation):
$$\tilde{m}(s)=\frac{\tilde{F}(s)}{1-\tilde{F}(s)}~~~~~~~~~~~~~~~~(1)$$ where $\tilde{F}(s)$ is the LST of $F(t)$, i.e., $\tilde{F}(s)=\int_0^\infty e^{-st}\mathrm dF(t)=\int_0^\infty e^{-st}f(t)\mathrm dt$.
Here comes the question, actually the LST of $F(t)$ does not exist due to the $\frac{1}{t^{\alpha+1}}$ form in $f(t)$. So how can I compute $\tilde{m}(s)$ according to Formula (1)?
I have been stuck on this question for several days, your help is very much appreciated!