## FANDOM

33.178 Seiten

 Typus Verschleierung Bearbeiter Graf Isolan Gesichtet
Untersuchte Arbeit:
Seite: 15, Zeilen: 1-10
Quelle: Degen 2006
Seite(n): 5, Zeilen: 13-23
Theorem 2.2.5. (Characterization of MDAα(x)))

The distribution function F belongs to the maximum domain of attraction of Φα(x) (α>0) if and only if ${\scriptstyle \bar{F}(x)\in \mathcal{R}_{-\alpha}}$. In that case

an-1Mn $\xrightarrow{d}$ Φα

with ${\scriptstyle a_n = F^\leftarrow(1-1/n),}$ where ${\scriptstyle F^\leftarrow}$ is the quantile function.

Proof. See for instance Resnick (1987), Proposition 1.11, pp. 54. ${\scriptstyle\square}$

By Taylor expansion, we can see that 1-Φα(x)= 1-exp(-x) ~ x as ${\scriptstyle x\to\infty,}$ i.e., the tail of Φα decreases like a power law. At this point is where the concept of regular variation play [sic!] an important role. Indeed, regular variation tells us how far we can move from exact power law behaviour and still remain in MDAα(x)).

By Taylor expansion, 1-Φα(x)= 1-exp(-x) ~ x as ${\scriptstyle x\to\infty,}$ i.e. the tail of Φα decreases like a power law. At this point the concept of regular variation enters; see Appendix C. Regular variation tells us how far we can move from exact power law behavior and still remain in MDAα(x)). The following theorem traces back to Gnedenko (1943):

Theorem 1.2 (Characterization of MDAα(x)))

The df F belongs to the maximum domain of attraction of Φα(x) (α>0) if and only if ${\scriptstyle\bar{F}(x)\in RV_{-\alpha} }$. In that case

an-1Mn ${\scriptstyle\xrightarrow{d}}$ Φα

with ${\scriptstyle a_n = F^\leftarrow(1-1/n).}$

Proof. See for instance Resnick [34], pp. 54-57.${\scriptstyle \square}$

 Anmerkungen Keine Kennzeichnung der Übernahme, keine Quellenangabe. Wo Rh von der Vorlage abweicht, finden sich Grammatikfehler. Die Herkunft des Theorems, welche von Degen explizit gemacht wurde, wird bei Rh verschwiegen. Sichter (Graf Isolan), KnallErbse