西南大学学报 (自然科学版)  2019, Vol. 41 Issue (1): 78-83.  DOI: 10.13718/j.cnki.xdzk.2019.01.012 0
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 $\mathop {\lim }\limits_{n \to \infty } \mathbb{P}\left( {{M_x} \leqslant {\alpha _n}{{\left| x \right|}^{{\beta _n}}}{\text{sign}}\left( x \right)} \right) = \mathop {\lim }\limits_{n \to \infty } {F^n}\left( {{\alpha _n}{{\left| x \right|}^{{\beta _n}}}{\text{sign}}\left( x \right)} \right) = H\left( x \right)$ (1)

 ${\rm{I}}:{H_{1,\alpha }}\left( x \right) = \left\{ {\begin{array}{*{20}{c}} \begin{array}{l} 0\\ \exp \left( { - {{\left( {\log x} \right)}^{ - \alpha }}} \right) \end{array}&\begin{array}{l} x \le 1\\ x > 1 \end{array} \end{array}} \right.$
 ${\rm{II}}:{H_{2,\alpha }}\left( x \right) = \left\{ \begin{array}{l} 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \le 0\\ \exp \left( { - {{\left( { - \log x} \right)}^\alpha }} \right)\;\;\;\;\;\;\;0 < x < 1\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge 1 \end{array} \right.$
 ${\rm{III}}:{H_{3,\alpha }}\left( x \right) = \left\{ \begin{array}{l} 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \le - 1\\ \exp \left( { - {{\left( { - \log \left( { - x} \right)} \right)}^{ - \alpha }}} \right)\;\;\;\;\;\;\; - 1 < x < 0\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge 0 \end{array} \right.$
 ${\rm{IV}}:{H_{4,\alpha }}\left( x \right) = \left\{ \begin{array}{l} \exp \left( { - {{\left( {\log \left( { - x} \right)} \right)}^\alpha }} \right)\;\;\;\;\;\;x \le - 1\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge - 1 \end{array} \right.$
 ${\rm{V}}:\mathit{\Phi }\left( x \right) = \left\{ \begin{array}{l} 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \le 0\\ \exp \left( { - \frac{1}{x}} \right)\;\;\;\;\;\;\;\;x > 0 \end{array} \right.$
 ${\rm{VI}}:\mathit{\Psi }\left( x \right) = \left\{ \begin{array}{l} \exp \left( x \right)\;\;\;\;\;x < 0\\ 1\;\;\;\;\;\;\;\;\;\;\;\;\;\;x \ge 0 \end{array} \right.$

 $U\left( t \right) = \int_t^\infty {u\left( s \right){\rm{d}}s}$

 $\mathop {\lim }\limits_{t \to \infty } \frac{{tu\left( t \right)}}{{U\left( t \right)}} = \rho$ (2)

 ${\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| X \right|} \right) - t} \right)}^p}\left| {\left( {\log \left| X \right|} \right) > t} \right.} \right\}$

 $\frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( {\alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( {\alpha - p - 2} \right)}}$ (3)

关于必要性：由文献[2]之定理2.1知，FDp(H1，α)的充要条件为对于y＞0时，

 $\mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( {\exp \left( {ty} \right)} \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}} = {y^{ - \alpha }}$

α＞2时有

 $\begin{array}{l} {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| X \right|} \right) - t} \right)}^p}\left| {\left( {\log \left| X \right|} \right) > t} \right.} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;\int_{\exp \left( t \right)}^\infty {\frac{{{{\left( {\left( {\log \left| x \right|} \right) - t} \right)}\;^p}{\rm{d}}F\left( x \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}}} = p{t^p}\int_1^\infty {{{\left( {u - 1} \right)}^{p - 1}}\frac{{1 - F\left( {\exp \left( {tu} \right)} \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}}{\rm{d}}u} \to \\ \;\;\;\;\;\;\;\;\;\;\;p{t^p}\int_1^\infty {{{\left( {u - 1} \right)}^{p - 1}}{u^{ - \alpha }}{\rm{d}}u} ,\alpha > 0 = \frac{{\mathit{\Gamma }\left( {\alpha - p} \right)\mathit{\Gamma }\left( {p + 1} \right)}}{{\mathit{\Gamma }\left( \alpha \right)}}{t^p} \end{array}$ (4)

 ${J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{\exp \left( t \right)}^\infty {{{\left( {\left( {\log \left| x \right|} \right) - t} \right)}^p}{\rm{d}}F\left( x \right)}$

 $\begin{array}{l} \int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( u \right){\rm{d}}u} = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_t^\infty {\int_{\exp \left( u \right)}^\infty {{{\left( {u - t} \right)}^\delta }{{\left( {\left( {\log \left| x \right|} \right) - u} \right)}^p}{\rm{d}}F\left( x \right){\rm{d}}u} } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{\exp \left( t \right)}^\infty {\int_t^{\log \left| x \right|} {{{\left( {u - t} \right)}^\delta }{{\left( {\left( {\log \left| x \right|} \right) - u} \right)}^p}{\rm{d}}u{\rm{d}}F\left( x \right)} } = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\frac{{\mathit{\Gamma }\left( {\delta + 1} \right)}}{{\mathit{\Gamma }\left( {p + \delta + 2} \right)}}\int_{\exp \left( t \right)}^\infty {{{\left( {\left( {\log \left| x \right|} \right) - t} \right)}^{p + \delta }}{\rm{d}}F\left( x \right)} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( t \right) \end{array}$ (5)

 ${J_{p + 1}}\left( t \right) = \int_t^\infty {{J_p}\left( u \right){\rm{d}}u} \downarrow 0$ (6)

 ${\mu _p}\left( t \right) = \frac{{\mathit{\Gamma }\left( {p + 1} \right)}}{{1 - F\left( {\exp \left( t \right)} \right)}}{J_p}\left( t \right)$

 $\frac{{J\left( t \right){J_{p + 2}}\left( t \right)}}{{{{\left\{ {{J_{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\alpha - p - 1}}{{\alpha - p - 2}}$ (7)

 $\begin{array}{l} {\left( {{J_0}^ * \left( t \right)} \right)^\prime } = {\left( {\frac{{J_1^2\left( t \right)}}{{{J_2}\left( t \right)}}} \right)^\prime } = \frac{{{J_1}^3\left( t \right)}}{{{J_2}^2\left( t \right)}}\left( {1 - \frac{{2{J_0}\left( t \right){J_2}\left( t \right)}}{{{J_1}^2\left( t \right)}}} \right) = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; - \frac{1}{t}{J_0}^ * \left( t \right)\frac{t}{{f\left( t \right)}}\left( {2\frac{{{J_0}\left( t \right)}}{{{J_0}^ * \left( t \right)}} - 1} \right) \end{array}$

 ${J_p}^ * \left( t \right) = \frac{{{{\left( {{J_{p + 1}}\left( t \right)} \right)}^2}}}{{{J_{p + 2}}\left( t \right)}}$

 $\frac{{{J_p}\left( t \right)}}{{{J_p}^ * \left( t \right)}} \to \frac{{\alpha - p - 1}}{{\alpha - p - 2}}$

p=0的方法相同，很容易得到Jp(t)∈RV-(α-p).由式(6)和引理1，得JP-1RV-(α-p)-1.将上述步骤重复p次，就得到1-F(exp(t))=J0(t)∈RV-α.

 ${J_{\delta + p + 1}}\left( t \right) = \frac{1}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( u \right){\rm{d}}u} = \frac{{{t^{\delta + 1}}}}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_t^\infty {{{\left( {y - 1} \right)}^\delta }{J_p}\left( {ty} \right){\rm{d}}y}$

 $\begin{array}{*{20}{c}} {\frac{{{J_{\delta + p + 1}}\left( {tx} \right)}}{{{t^{\delta + 1}}{J_p}\left( t \right)}} = \frac{{{x^{\delta + 1}}}}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_1^\infty {{{\left( {y - 1} \right)}^\delta }\frac{{{J_p}\left( {txy} \right)}}{{{J_p}\left( t \right)}}{\rm{d}}y} \to }\\ {\frac{{{x^{ - \alpha + p + 1 + \delta }}}}{{\mathit{\Gamma }\left( {\delta + 1} \right)}}\int_1^\infty {{{\left( {y - 1} \right)}^\delta }{y^{ - \alpha + p}}{\rm{d}}y} = }\\ {{x^{ - \alpha + p + 1 + \delta }}\frac{{\mathit{\Gamma }\left( {\alpha - p - \delta - 1} \right)}}{{\mathit{\Gamma }\left( {\alpha - p} \right)}}} \end{array}$

 $1 - F\left( {\exp \left( t \right)} \right) = {J_0}\left( t \right) \in R{V_{ - \alpha }}$

 ${\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) - \frac{1}{t}} \right)}^p}\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) > \frac{1}{t}} \right\}$

 $\frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( { - \alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( { - \alpha - p - 2} \right)}}$ (8)

必要性相的证明似于定理1中(4)的证明，利用文献[2]中定理2.2给出的FDp(H2α)充要条件为当y＞0时，

 $\mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( {r\left( F \right)\exp \left( { - \frac{y}{t}} \right)} \right)}}{{1 - F\left( {r\left( F \right)\exp \left( { - \frac{1}{t}} \right)} \right)}} = {y^\alpha }$

 ${J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{r\left( F \right)\exp \left( { - \frac{1}{t}} \right)}^{r\left( F \right)} {{{\left( {\left( { - \log \left| {\frac{x}{{r\left( F \right)}}} \right|} \right) - \frac{1}{t}} \right)}^p}{\rm{d}}F\left( x \right)}$

 $\int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( {\frac{1}{u}} \right){\rm{d}}u} = \mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {\frac{1}{t}} \right)$ (9)

 ${\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| { - X} \right|} \right) + t} \right)}^p}\left| {\left( {\log \left| { - X} \right|} \right) > - t} \right.} \right\}$

 $\frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( { \alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( { \alpha - p - 2} \right)}}$ (10)

必要性的证明类似于定理1中(4)式的证明，利用文献[2]中定理2.3关于FDp(H3α)的充要条件为当y＞0时，

 $\mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( { - \exp \left( { - ty} \right)} \right)}}{{1 - F\left( { - \exp \left( { - t} \right)} \right)}} = {y^{ - \alpha }}$

 ${J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{ - \exp \left( { - t} \right)}^0 {{{\left( {\left( {\log \left| { - X} \right|} \right) + t} \right)}^p}{\rm{d}}F\left( x \right)}$

 $\int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( u \right){\rm{d}}u} = {\left( { - 1} \right)^{\delta + 1}}\mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( t \right)$ (11)

 ${\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) + \frac{1}{t}} \right)}^p}\left| {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) > - \frac{1}{t}} \right.} \right\}$

 $\frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{\left( {p + 2} \right)\left( { - \alpha - p - 1} \right)}}{{\left( {p + 1} \right)\left( { - \alpha - p - 2} \right)}}$ (12)

必要性的证明类似于定理1中(4)式的证明，利用文献[2]中定理2.4关于FDp(H4α)的充要条件为当y＞0时

 $\mathop {\lim }\limits_{t \to \infty } \frac{{1 - F\left( {r\left( F \right)\exp \left( {\frac{y}{t}} \right)} \right)}}{{1 - F\left( {r\left( F \right)\exp \left( {\frac{1}{t}} \right)} \right)}} = {y^\alpha }$

 ${J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{r\left( F \right)\exp \left( {\frac{1}{t}} \right)}^{r\left( F \right)} {{{\left( {\left( { - \log \left| {\frac{X}{{r\left( F \right)}}} \right|} \right) + \frac{1}{t}} \right)}^p}{\rm{d}}F\left( x \right)}$

 $\int_t^\infty {{{\left( {u - t} \right)}^\delta }{J_p}\left( {\frac{1}{u}} \right){\rm{d}}u} = {\left( { - 1} \right)^{\delta + 1}}\mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {\frac{1}{t}} \right)$ (13)

 ${\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) - f\left( t \right)} \right)}^p}\left| {\left( {\log \left| {\frac{X}{t}} \right|} \right) > f\left( t \right)} \right.} \right\}$

 $\frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{p + 2}}{{p + 1}}$ (14)

必要性的证明：由文献[2]定理2.5可知，FDp(Φ)的等价条件为当r(F)＞0时，存在一个函数f＞0，使得对于x$\mathbb{R}$

 $\mathop {\lim }\limits_{t \to r\left( F \right)} \frac{{1 - F\left( {t\exp \left( {xf\left( t \right)} \right)} \right)}}{{1 - F\left( t \right)}}\exp \left( { - x} \right)$

 $f\left( t \right) = \frac{1}{{\bar F\left( t \right)}}\int_t^{r\left( F \right)} {\frac{{\bar F\left( s \right)}}{s}{\rm{d}}s}$

 $\int_t^{r\left( F \right)} {\frac{{\bar F\left( x \right)}}{x}{\rm{d}}x} < \infty$

 $\begin{array}{l} {\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) - f\left( t \right)} \right)}^p}\left| {\left( {\log \left| {\frac{X}{t}} \right|} \right) > f\left( t \right)} \right.} \right\} = \\ \;\;\;\;\;\;\;\;\;\;\;p{\left( {f\left( t \right)} \right)^p}\int_1^{\frac{{\log \left| {\frac{{r\left( F \right)}}{t}} \right|}}{{f\left( t \right)}}} {\frac{{1 - F\left( {t\exp \left( {yf\left( t \right)} \right)} \right)}}{{1 - F\left( {t\exp \left( {f\left( t \right)} \right)} \right)}}{{\left( {y - 1} \right)}^{p - 1}}{\rm{d}}y} \sim \\ \;\;\;\;\;\;\;\;\;\;\;\mathit{\Gamma }\left( {p + 1} \right){\left( {f\left( t \right)} \right)^p},t \uparrow r\left( F \right) \end{array}$ (15)

f(t)的表达式可知

 $\frac{{\log \left| {\frac{{r\left( F \right)}}{t}} \right|}}{{f\left( t \right)}} - 1 \to \infty \left( {t \uparrow r\left( F \right)} \right)$

 ${J_p}\left( t \right) = {\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}^{ - 1}}\int_{t\exp \left( {f\left( t \right)} \right)}^{r\left( F \right)} {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) - f\left( t \right)} \right)}^p}{\rm{d}}F\left( x \right)} \;\;\;\;\;\;\;\;\;\;t < r\left( F \right)$

 ${g_1}\left( t \right) = \log \left( t \right) + f\left( t \right)$

 $\int_t^\infty {{{\left( {u - {g_1}\left( t \right)} \right)}^\delta }{J_p}\left( {g_1^ \leftarrow \left( u \right)} \right){\rm{d}}u} = \mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {{g_1}\left( t \right)} \right)$ (16)

 ${\mu _p}\left( t \right) = E\left\{ {{{\left( {\left( {\log \left| {\frac{X}{t}} \right|} \right) + f\left( t \right)} \right)}^p}\left| {\left( {\log \left| {\frac{X}{t}} \right|} \right) > - f\left( t \right)} \right.} \right\}$

 $\frac{{{\mu _p}\left( t \right){\mu _{p + 2}}\left( t \right)}}{{{{\left\{ {{\mu _{p + 1}}\left( t \right)} \right\}}^2}}} \to \frac{{p + 2}}{{p + 1}}$ (17)

必要性的证明：由文献[2]定理2.6可知，FDp(Ψ)的等价条件为当r(F)≤0时，存在一个函数f＞0使得对于x$\mathbb{R}$，有

 $\mathop {\lim }\limits_{t \to r\left( F \right)} \frac{{1 - F\left( {t\exp \left( {xf\left( t \right)} \right)} \right)}}{{1 - F\left( t \right)}} = \exp \left( x \right)$

 $f\left( t \right) = - \frac{1}{{\bar F\left( t \right)}}\int_t^{r\left( F \right)} {\frac{{\bar F\left( s \right)}}{s}{\rm{d}}s}$

 $- \int_t^{r\left( F \right)} {\frac{{\bar F\left( x \right)}}{x}{\rm{d}}x} < \infty$

 $\begin{array}{*{20}{c}} {{\mu _p}\left( t \right) \sim \mathit{\Gamma }\left( {p + 1} \right){{\left( { - f\left( t \right)} \right)}^p}}&{r \uparrow r\left( F \right)} \end{array}$ (18)

 $\begin{array}{*{20}{c}} {{J_p}\left( t \right) = {{\left\{ {\mathit{\Gamma }\left( {p + 1} \right)} \right\}}^{ - 1}}\int_{t\exp \left( { - f\left( t \right)} \right)}^{r\left( F \right)} {{{\left( {\left( {\log \left| {\frac{x}{t}} \right|} \right) + f\left( t \right)} \right)}^p}{\rm{d}}F\left( x \right)} }&{t < r\left( F \right)} \end{array}$

 $\int_t^\infty {{{\left( {u - {g_2}\left( t \right)} \right)}^\delta }{J_p}\left( {g_2^ \leftarrow \left( u \right)} \right){\rm{d}}u} = \mathit{\Gamma }\left( {\delta + 1} \right){J_{\delta + p + 1}}\left( {{g_2}\left( t \right)} \right)$ (19)

 [1] PANTCHEVA E. Limit Theorems for Extreme Order Statistics Under Nonlinear Normalization[M]//KALASHNIKOV V V, ZOLOTATEV V M. Stability Problems for Stochastic Models. Berlin: Springer, 1985: 284-309. [2] MOHAN N R, RAVI S. Max Domains of Attraction of Univariate and Multivariate P-Max Stable Laws[J]. Theory of Probability and Its Applications, 1993, 37(4): 632-643. DOI:10.1137/1137119 [3] PENG Z X, SHUAI Y L, NADARAJAH S. On Convergence of Extremes Under Power Normalization[J]. Extremes, 2013, 16(3): 285-301. DOI:10.1007/s10687-012-0161-2 [4] RESNICK S I. Extreme Values, Regular Variation, and Point Processes[M]. New York: Springer-Verlag, 1987. [5] GELUK J L. On The Domain of Attraction of exp (-exp (-x))[J]. Statistics and Probability Letters, 1996, 31(2): 91-95.
Conditions Based on Conditional Moments for p-Max Stable Laws Conditional Moment Characterization of Limit Distribution Under Power Normalization
PENG Xi, ZHOU Wei, PENG Zuo-xiang
School of Mathematics and Statistics, Southwest University, Chongqing 400715, China
Abstract: Let {Xn} be independent and identically distributed random variables with the common distribution function F(x). Necessary and sufficient conditions for F belonging to the domains of attraction of H1, α, H2, α, H3, α, H4, α, ψ(x) and Ф(x) with nondegenerate univariate marginals under power normalization are derived in terms of conditional moments.
Key words: limit distribution under power normalization    conditional moment    domain of attraction