Timeline for convergence in distribution and convergence of moments
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 23, 2015 at 16:29 | vote | accept | S. W. M | ||
| Jan 23, 2015 at 14:51 | comment | added | S. W. M | In addition, I've already checked the convergence of three first moments. The combination of these facts made me hopeful about the convergence of all moments, at least in this situation. Thanks for the reply. | |
| Jan 23, 2015 at 14:28 | comment | added | S. W. M | I understand your argument. In general, we can't affirm the convergence in $(K + 1)$-th order moment from the convergence of $K$-th order moment. I've been wondering about the Theorem 25.12's Corollary in Billingsley's book (Probability and Measure, English version, third edition, 1995, page 338), which requires $\sup_n[|X_n|^{k+\varepsilon}]<\infty$ plus convergence in distribution to get the $K$-th order moment convergence. In my case, the $X_n$'s Moment generating function exists in $|t|<t_0$ for each $n$. | |
| Jan 23, 2015 at 9:09 | comment | added | Ori Gurel-Gurevich | This also works, mutatis mutandis, for any K and any distribution. | |
| Jan 22, 2015 at 23:55 | history | answered | Nate Eldredge | CC BY-SA 3.0 |