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I am now very busy to preparing my stochastic modelling exam and I feel sad when dealing with some problems.

1.For a renewal process $ {N(t):t∈[0,∞)} $, how to get the limiting distribution of the total lifetime $\beta_t$? $\beta_t$ is defined as $\beta_t = γ_t + δ_t$ , where $γ_t$ is the residual lifetime and $δ_t$ is the current lifetime. I have got the limiting joint c.d.f of $γ_t$ and $δ_t$, where $$P(γ_t>x,δ_t>y )→\frac{1}\mu\int_{x+y}^{\infty}(1-F(s))ds$$,as $t→\infty.$ $F(s)$ is the c.d.f of the inter-occurrence time random variable and $\mu$ is the mean of the inter-occurrence time random variable.

2.The inter-occurrence times of a renewal process $ {N(t):t∈[0,∞)} $ are iid following the uniform distribution on $[0,1]$. The limiting distribution of the excess life time $γ_t$ is such that, as $t→∞$,$$P(γ_t≤x)→2\int_0^x(1−s)ds$$ for $x∈[0,1]$. Based on this fact, derive $$ \lim_{x→∞} P(γ_t>1/2,δ_t<1/3) $$where $δ_t$ is the current life time.

For the second question, I have ask my instructor and he told me that just directly calculate it by the above joint c.d.f but I really have no idea on it. The answer is $\frac{1}3$. Does anyone can help me solve these 2 problems? All the notations follow the textbook "An Introduction to Stochastic Modeling" by Taylor and Karlin. Many thanks.

Garvin Cheng Hok Laam
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  • Re 1., differentiating twice the joint PDF you were given yields the joint PDF of $(\gamma,\delta)$ as $$\mu^{-1}f(x+y)\mathbf 1_{x>0,y>0}$$ hence the PDF of $\beta=\gamma+\delta$ is $$f_\beta(z)=\int_0^z\mu^{-1}f(z)dx=\mu^{-1}zf(z)$$ on $z>0$. Re 2., you are asked to note that $$P(\gamma>x,\delta<y)=P(\gamma>x)-P(\gamma>x,\delta>y)=1-P(\gamma\leqslant x)-P(\gamma>x,\delta>y)$$ and that you know both terms on the RHS. – Did May 14 '17 at 08:11
  • Many thanks, but I wonder how to get the pdf when the cdf is an improper integral? – Garvin Cheng Hok Laam May 15 '17 at 04:17
  • The PDF needs not be defined at every point. – Did May 15 '17 at 06:10

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