On the extreme-value theory for stationary diffusions under power normalization (Q1881755)

For \(a,b>0\) let \(p_{a,b}(x)=a| x| ^{b}\text{sgn}\,x\), \(x\in {\mathbb R}\). For \(F\), \(Q\) probabilities on \({\mathbb R}\), \(Q\) nondegenerate, \(F\in MDA_{p}(Q)\) means \(Q=\lim F\circ p_{a_{n},b_{n}}^{-1}\) for some \(a_{n}\), \(b_{n}\); \(Q\) must be \(p\)-max stable, i.e., for every \(n\), \(Q^{n}=Q\circ p_{a_{n},b_{n}}^{-1}\) for some \(a_{n}\), \(b_{n}\), where \(Q^{n}((-\infty ,x])=Q((-\infty ,x])^{n}\). The max-stable distribution functions are (besides a \(\circ p_{a,b}^{-1}\)), \(H_{1,\alpha }=\Phi_{\alpha }(\log (\max (x,1)))\), \(H_{2,\alpha }(x)=\Psi_{\alpha }(\log (\max (x,0)))\), \(H_{3,\alpha }(x) = \Phi_{\alpha }(-\log (\max (-x,0)))\), \(H_{4,\alpha }(x)=\Psi_{\alpha }(-\log (\max (-x,0)))\), \(H_{5}=\Phi_{1}\), \(H_{6}=\Psi_{1}\), where \(\Phi_{\alpha }(x)=\exp (-\max (x,0)^{-\alpha })\), \(\Psi_{\alpha }(x)=\exp (-\max (-x,0)^{\alpha })\), \(\alpha >0\). For a distribution function \(F(x)\) let \(r(F)=\sup\{x;F(x)<1\}\), \(F_{1}(x)=1-F(x)\). If the derivative \(f= F'\) exists on an \((x_{0},r(F))\), the author establishes sufficient conditions for \(F\in MDA_{p}(H_{i,\alpha })\), \(MDA_{p}(H_{j})\), \(i=1,\dots,4\), \(j=5,6\). Namely: \(i=1\), \(r(F)=\infty\), \(\lim _{x\uparrow r(F)}f(x)x\log x/F_{1}(x)=\alpha\); \(i=2\), \(r(F)\in (0,\infty )\), \(\lim _{x\uparrow r(F)}f(x)x\log (r(F)/x)/F_{1}(x)=\alpha\); \(i=3\), \(r(F)=0\), \(\lim _{x\uparrow 0}f(x)x\log | x| /F_{1}(x)= \alpha\); \(i=4\), \(r(F)<0\), \(\lim _{x\uparrow r(F)} f(x)| x| \log (x/r(F))/F_{1}(x)=\alpha\); \(j=5,6\), \(f'\) exists, \(\lim _{x\uparrow r(F)} [(F_{1}(x)/f(x)x)+(F_{1}(x)f'(x)/f(x)^{2})]=-1\), \(r(F)\in (0,\infty ]\) for \(j=5\) and \(r(F)\leq 0\) for \(j=6\). They are deduced from necessary and sufficient conditions in terms of \(F\), proved by relying on those for \(x\rightarrow a x + b\), \(a>0\), instead of \(p_{a,b}\). Then the author considers a regular diffusion \(X(t)\) on \((q,r)\subset {\mathbb R}\), with scale \(s\), \(s((q,r))={\mathbb R}\), and finite speed measure \(m\), \(| m| =m((q,r))\), and \(M_{T}=\) max\(_{0\leq t\leq T}X(t)\). Let \({\mathcal R}_{\alpha }(a)\) be the class of functions with regular variation of order \(\alpha\) in \(a\). Using the fact that for the distribution function \(G\) of \(M_{T}\), there exist \(u_{T}\uparrow r\) for \(T\rightarrow\infty\), such that \(G(u_{T}) -\exp(-T/s(u_{T})| m| )\rightarrow 0\), analogous necessary and sufficient conditions for \(F\in MDA_{p}(H)\) in the six cases are deduced, expressed as the previous ones but with \(r\) instead of \(r(F)\), and with the limit relations replaced by \(s(e^{\cdot })\in {\mathcal R}_{\alpha }(\log r)\) for \(i=1,2\), \(s(-e^{-\cdot })\in {\mathcal R}_{\alpha }(-\log | r| )\) for \(i=3,4\), and by the existence of an \(a(y)>0\) such that \(\lim _{y\uparrow r}s(y e^{a(y)x})/s(y)=e^{\pm x}\), \(x\in (x_{0},r)\) for \(j=5,6\), with \(r>0\) and sign \(+\) for \(j= 5\) and \(r\leq 0\) and sign \(-\) for \(j=6\). In all the 12 situations, necessary and sufficient conditions are accompanied by expressions of \(a_{n}\), \(b_{n}\). Sufficient conditions, in terms of \(s '\) (supposed to exist) instead of \(f '\), are deduced. Remarks concerning min instead of max and translations for solutions of Itô equations are stated.