Timeline for Determine the affine envelope of a random process's MGF
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 24, 2021 at 1:12 | comment | added | leeyee | Your solutions to my original question are very useful. And thanks for reminding me the policy. I have posted the further question as a separate one on MSE: math.stackexchange.com/questions/4340872/…. | |
| Dec 23, 2021 at 22:25 | vote | accept | leeyee | ||
| Dec 23, 2021 at 14:42 | comment | added | Iosif Pinelis | This limit is just $1$ for all $x$ such that $P(B_1\le x)<1$. However, the questions in your latter comment are in addition to your posted question. You may want to post such additional questions separately (not necessarily on MathOverflow). Generally, asking multiple questions in one post is discouraged on MathOverflow. Anyway, are you satisfied with the answers to your originally posted question? | |
| Dec 23, 2021 at 10:25 | comment | added | leeyee | The bound of (*) seems to be slightly tighter in my application. As all I need is a lower envelope of $S(t)$, I wonder is that possible to directly derive an upper bound of $\lim_{n\rightarrow\infty}P(\max_{i=1,\ldots,n}B_i>x)$, if I know $B$'s MGF exactly? As an example, suppose that $B$ is the busy period of a M/M/1 queue, for which the Laplace transform of its pdf is known to be $G^*(s)=\frac{\mu+\lambda+s-[(\mu+\lambda+s)^2-4\lambda\mu]^{1/2}}{2\lambda}$, where $\lambda$ and $\mu$ are arrival and service rates. respectively [Eqn.(5.144) of Kleinrock "Queueing Systems, Vol. 1"]. | |
| Dec 23, 2021 at 2:42 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |