Asymptotics

Abstract

Now let us recall main results on limit theorems for record values in the sequences of i.i.d. random variables $\{X_n,n\ge 1\}$ with a common continuous distribution function $F$ which will be applied to the case of Student’s $t$-distributions.

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2.1 Asymptotic Behavior of Student’s Pdf

Proposition 2.1

For each$x\in R^d$, as$\nu \rightarrow \infty$,

$$\begin{aligned} f_{\nu ,\Sigma ,a}(x)\rightarrow g_{a,\Sigma }(x). \end{aligned}$$

(2.1)

Proof

Let $a=0$. Using the well-known formula that

$$\begin{aligned} \Gamma (z)=\sqrt{\frac{2\pi }{z}}e^{-z}z^z\left(1+O\left(\frac{1}{z}\right)\right), \quad \mathrm as \quad z\rightarrow \infty , \end{aligned}$$

(2.2)

we find that, as $\nu \rightarrow \infty$,

$$\begin{aligned} \frac{\Gamma (\frac{\nu +d}{2})}{(\nu \pi )^{\frac{d}{2}}\Gamma (\frac{\nu }{2})}\sim \frac{\sqrt{\frac{4\pi }{\nu +d}}e^{-\frac{\nu +d}{2}}(\frac{\nu +d}{2})^{\frac{\nu +d}{2}}}{(\nu \pi )^{\frac{d}{2}}\sqrt{\frac{4\pi }{\nu }}e^{-\frac{\nu }{2}}(\frac{\nu }{2})^{\frac{\nu }{2}}}\rightarrow \frac{1}{(2\pi )^{\frac{d}{2}}} \end{aligned}$$

(2.3)

and, obviously,

$$\begin{aligned} \left(1+\frac{\langle x\Sigma ^{-1},x\rangle }{\nu }\right)^{-\frac{\nu +d}{2}}\rightarrow e^{-\frac{1}{2}\langle x\Sigma ^{-1},x\rangle }. \end{aligned}$$

(2.4)

Here and below “$\sim$” is the equivalence sign.

The statement (2.1) with $a=0$ follows from (1.1), (2.2), (2.3) and (2.4).

Let now $a\ne 0$ and

$$\begin{aligned} y_\nu =\frac{2}{\nu +d}\left[\langle a\Sigma ^{-1},a\rangle (\nu +\langle x\Sigma ^{-1},x\rangle )\right]^{\frac{1}{2}}. \end{aligned}$$

Because, as $\nu \rightarrow \infty$, uniformly in $y$ (see Appendix)

$$\begin{aligned} K_{\nu }(\nu y)\sim \sqrt{\frac{\pi }{2\nu }}\frac{\exp {\{-\nu \sqrt{1+y^2}\}}}{(1+y^2)^{\frac{1}{4}}}\left(\frac{y}{1+\sqrt{1+y^2}}\right)^{-\nu } \end{aligned}$$

and

$$\begin{aligned} \sqrt{1+y_{\nu }^2}\sim 1+\frac{1}{2}y_{\nu }^2, \end{aligned}$$

we shall have that

$$\begin{aligned}&K_{\frac{\nu +d}{2}}\left(\left[\langle a\Sigma ^{-1},a\rangle (\nu +\langle x\Sigma ^{-1},x\rangle )\right]^{\frac{1}{2}}\right) =K_{\frac{\nu +d}{2}}\left(\frac{\nu +d}{2}y_{\nu }\right) \nonumber \\&\sim \sqrt{\frac{\pi }{\nu +d}}\exp \left\{ -\frac{\nu +d}{2}\left(1+\frac{1}{2}y_{\nu }^2\right)\right\} \left(\frac{y_{\nu }}{2+\frac{1}{2}y_{\nu }^2}\right)^{-\frac{\nu +d}{2}} \nonumber \\&\sim \sqrt{\frac{\pi }{\nu +d}}e^{-\frac{\nu +d}{2}}\exp {\left\{ -\frac{1}{\nu +d}\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle x\Sigma ^{-1},x\rangle \right)\right\} }\left(\frac{y_{\nu }}{2+\frac{1}{2}y_{\nu }^2}\right)^{-\frac{\nu +d}{2}}.\nonumber \\ \end{aligned}$$

(2.5)

From (1.2) and (2.5) we elementarily find that

$$\begin{aligned} f_{\nu ,\Sigma ,a}(x)&\sim \frac{(\frac{\nu }{2})^{\frac{\nu }{2}}}{\Gamma (\frac{\nu }{2})}\frac{2\exp \left\{ \langle x \Sigma ^{-1},a\rangle \right\} }{(2\pi )^{\frac{d}{2}}\sqrt{|\Sigma |}}\left(\frac{\langle a\Sigma ^{-1},a\rangle }{\nu _+\langle x\Sigma ^{-1},x}\right)^{\frac{\nu +d}{4}}\sqrt{\frac{\pi }{\nu +d}}e^{-\frac{\nu +d}{2}} \nonumber \\&\quad \times \exp \left\{ -\frac{1}{\nu +d}\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle x\Sigma ^{-1},x\rangle \right)\right\} \left(\frac{y_{\nu }}{2+\frac{1}{2}y_{\nu }^2}\right)^{-\frac{\nu +d}{2}} \nonumber \\&\sim \frac{\exp \left\{ \langle x\Sigma ^{-1},a\rangle \right\} }{(2\pi )^{\frac{d}{2}}\sqrt{|\Sigma |}}e^{-\langle a\Sigma ^{-1},a\rangle }e^{-\frac{d}{2}}\left(\frac{\nu +\langle x\Sigma ^{-1},x\rangle }{2+\frac{1}{2}y_{\nu }^2}\right)^{-\frac{\nu +d}{2}} \nonumber \\&\sim \frac{\exp \left\{ \langle x\Sigma ^{-1},a\rangle \right\} }{(2\pi )^{\frac{d}{2}}\sqrt{|\Sigma |}}e^{-\langle a\Sigma ^{-1},a\rangle }e^{-\frac{d}{2}}\nonumber \\&\quad \times \exp \left\{ -\frac{1}{2}(\langle x\Sigma ^{-1},x\rangle -d)\right\} \left(1+\frac{1}{4}y_{\nu }^2\right)^{\frac{\nu +d}{2}}. \end{aligned}$$

(2.6)

But

$$\begin{aligned} \left(1+\frac{1}{4}y_{\nu }^2\right)^{\frac{\nu +d}{2}}&=\left(1+\frac{1}{(\nu +d)^2}\left[\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle x\Sigma ^{-1},x\rangle \right)\right]\right)^{\frac{\nu +d}{2}}\nonumber \\&\rightarrow \exp \left\{ \frac{1}{2}\langle a\Sigma ^{-1},a\rangle \right\} . \end{aligned}$$

(2.7)

Thus, (2.6) and (2.7) imply that, for each $x\in R^d$, as $\nu \rightarrow \infty$,

$$\begin{aligned} f_{\nu ,\Sigma ,a}(x)\rightarrow \frac{\exp \left\{ \langle x\Sigma ^{-1},a\right\} }{(2\pi )^{\frac{d}{2}}\sqrt{|\Sigma |}}\exp \left\{ -\frac{1}{2}\left(\langle a\Sigma ^{-1},a\rangle +\langle x\Sigma ^{-1},x\rangle \right)\right\} =g_{a,\Sigma }(x). \;\, \square \end{aligned}$$

Proposition 2.2

For each fixed$x\epsilon R^d$and$\nu >0$, as$|a|\rightarrow 0$,

$$\begin{aligned} f_{\nu ,\Sigma ,a}(x)\rightarrow f_{\nu ,\Sigma }(x). \end{aligned}$$

Proof

Indeed, as $|a|\rightarrow 0$,

$$\begin{aligned}&K_{\frac{\nu +d}{2}}\left(\left[\langle a\Sigma ^{-1},a\rangle (\nu +\langle x\Sigma ^{-1},x\rangle )\right]^{\frac{1}{2}}\right)\\&\quad \sim \Gamma \left(\frac{\nu +d}{2}\right)2^{\frac{\nu +d}{2}-1}\left[\langle a\Sigma ^{-1},a\rangle (\nu +\langle x\Sigma ^{-1},x\rangle )\right]^{-\frac{\nu +d}{4}} \end{aligned}$$

(see Appendix) and, having in mind formulas (1.1), (1.2),

$$\begin{aligned} f_{\nu ,\Sigma ,a}(x)\rightarrow \frac{(\frac{\nu }{2})^{\frac{\nu }{2}}}{\Gamma (\frac{\nu }{2})}\frac{2^{\frac{\nu +d}{2}}\Gamma (\frac{\nu +d}{2})}{(2\pi )^{\frac{d}{2}}\sqrt{|\Sigma |}}\left(\nu +\langle x\Sigma ^{-1},x \rangle \right)^{-\frac{\nu +d}{2}}=f_{\nu ,\Sigma }(x).\qquad \qquad \qquad \quad \square \end{aligned}$$

Proposition 2.3

• (i) As$|x|\rightarrow \infty$,

  $$\begin{aligned} f_{\nu ,\Sigma }(x)\sim c_{\nu ,\Sigma }\left(\langle x\Sigma ^{-1},x\rangle \right)^{-\frac{\nu +d}{2}}, \end{aligned}$$

  where

  $$\begin{aligned} c_{\nu ,\Sigma }=\frac{\Gamma \left(\frac{d+\nu }{2}\right)}{\pi ^{\frac{d}{2}}\Gamma (\frac{\nu }{2})\sqrt{|\Sigma |}}. \end{aligned}$$

• (ii) As$|x|\rightarrow \infty$, $a\ne 0$,

  $$\begin{aligned} f_{\nu ,\Sigma ,a}(x)&\sim c_{\nu ,\Sigma ,a}\left(\langle x\Sigma ^{-1},x\rangle \right)^{-\frac{\nu +d+1}{4}}\\&\quad \times \exp {\left\{ -\left[\langle a\Sigma ^{-1},a\rangle \langle x\Sigma ^{-1},x\rangle \right]^{\frac{1}{2}}+\langle x\Sigma ^{-1},a\rangle \right\} }, \end{aligned}$$

  where

  $$\begin{aligned} c_{\nu ,\Sigma ,a}=\frac{(\frac{\nu }{2})^{\frac{\nu }{2}}\left(\langle a\Sigma ^{-1},a\rangle \right)^{\frac{\nu +d+1}{4}}}{\Gamma (\frac{\nu }{2})(2\pi )^{\frac{d-1}{2}}\sqrt{|\Sigma |}}. \end{aligned}$$

Proof

• (i) Obviously follows from (1.1).

• (ii) Because, as $|x|\rightarrow \infty$,

  $$\begin{aligned}&K_{\frac{\nu +d}{2}}\left(\left[\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle x\Sigma ^{-1},x\rangle \right)\right]^{\frac{1}{2}}\right) \nonumber \\&\quad \sim \sqrt{\frac{\pi }{2}}\left[\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle x\Sigma ^{-1},x\rangle \right)\right]^{-\frac{1}{4}}\nonumber \\&\qquad \times \exp \left\{ -\left[\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle x\Sigma ^{-1},x\rangle \right)\right]^{\frac{1}{2}}\right\} , \end{aligned}$$

  from (1.2) we find that, as $|x|\rightarrow \infty$,

  $$\begin{aligned} f_{\nu ,\Sigma ,a}(x)&\sim \frac{(\frac{\nu }{2})^{\frac{\nu }{2}}\left(\langle a\Sigma ^{-1},a\rangle \right)^{\frac{\nu +d-1}{4}}}{\Gamma (\frac{\nu }{2})(2\pi )^{\frac{d-1}{2}}\sqrt{|\Sigma |}}\frac{\exp {\left\{ \langle x\Sigma ^{-1},a\rangle \right\} }}{\left(\nu +\langle x\Sigma ^{-1},x\rangle \right)^{\frac{\nu +d+1}{4}}} \nonumber \\&\quad \times \exp \left\{ -\left[\langle a\Sigma ^{-1},a\rangle \left(\nu +\langle x\Sigma ^{-1},x\rangle \right)\right]^{\frac{1}{2}}\right\} \nonumber \\&\sim c_{\nu ,\Sigma ,a}\left(\langle x\Sigma ^{-1},x\rangle \right)^{-\frac{\nu +d+1}{4}}\nonumber \\&\quad \times \exp {\left\{ -\left[\langle a\Sigma ^{-1},a\rangle \langle x\Sigma ^{-1},x\rangle \right]^{\frac{1}{2}}+\langle x\Sigma ^{-1},a\rangle \right\} }.\qquad \qquad \qquad \quad \end{aligned}$$

  $\square$

Corollary 2.4

Let d=1.

• (i) If$a>0$, $x\rightarrow \infty$, then

  $$\begin{aligned} f_{\nu ,\sigma ^2,a}(x)\sim \frac{1}{\sigma \Gamma (\frac{\nu }{2})}\left(\frac{\nu a}{2\sigma }\right)^{\frac{\nu }{2}}x^{-\frac{\nu }{2}-1}. \end{aligned}$$

  (2.8)

• (ii) If$a>0$, $x\rightarrow -\infty$, then

  $$\begin{aligned} f_{\nu ,\sigma ^2,a}(x)\sim \frac{1}{\sigma \Gamma (\frac{\nu }{2})}\left(\frac{\nu a}{2\sigma }\right)^{\frac{\nu }{2}}|x|^{-\frac{\nu }{2}-1}\exp \left\{ -\frac{2a|x|}{\sigma ^2}\right\} . \end{aligned}$$

  (2.9)

• (iii) If$a<0$, $x\rightarrow \infty$, then

  $$\begin{aligned} f_{\nu ,\sigma ^2,a}(x)\sim \frac{1}{\sigma \Gamma (\frac{\nu }{2})}\left(\frac{\nu |a|}{2\sigma }\right)^{\frac{\nu }{2}}x^{-\frac{\nu }{2}-1}\exp \left\{ -\frac{2|a|x}{\sigma ^2}\right\} . \end{aligned}$$

  (2.10)

• (iv) If$a<0$, $x\rightarrow -\infty$, then

  $$\begin{aligned} f_{\nu ,\sigma ^2,a}(x)\sim \frac{1}{\sigma \Gamma (\frac{\nu }{2})}\left(\frac{\nu |a|}{2\sigma }\right)^{\frac{\nu }{2}}|x|^{-\frac{\nu }{2}-1}. \end{aligned}$$

  (2.11)

2.2 Asymptotic Distributions for Extremal and Record Values

Let now $d=1$ and $\{X_n, n\ge 1\}$ a sequence of i.i.d. random variables with common Student’s $t$-distribution function and let $M_n=\max \limits _{1\le j\le n}X_j$.

Proposition 2.5

• (i) If pdf of$\fancyscript{L}(X_1)$is$f_{\nu ,\sigma ^2}$, then, as$n\rightarrow \infty$,

  $$\begin{aligned} \fancyscript{L}\left((K_1n)^{-\frac{1}{\nu }}M_n\right)\Rightarrow \Phi _{\nu }, \end{aligned}$$

  where “$\Rightarrow$” means weak convergence of probability laws, $\Phi _{\nu }$ is the Fréchet distribution

  $$\begin{aligned} \Phi _{\nu }(x)=\left\{ \begin{array}{l} \exp {\left\{ -x^{-\nu }\right\} }, \quad \mathrm if \quad x>0 \\ 0, \quad \mathrm if \quad x\le 0, \end{array} \right. \end{aligned}$$

  and

  $$\begin{aligned} K_1=\frac{\Gamma (\frac{\nu +1}{2})\sigma ^{\nu }}{\nu \sqrt{\pi }\Gamma (\frac{\nu }{2})}. \end{aligned}$$

• (ii) If pdf of$\fancyscript{L}(X_1)$is$f_{\nu ,\sigma ^2,a}$, $a>0$, then, as$n\rightarrow \infty$,

  $$\begin{aligned} \fancyscript{L}\left((K_2n)^{-\frac{2}{\nu }}M_n\right)\Rightarrow \Phi _{\frac{\nu }{2}}, \end{aligned}$$

  where

  $$\begin{aligned} K_2=\frac{2(\frac{\nu a}{2\sigma })^{\frac{\nu }{2}}}{\nu \sigma \Gamma (\frac{\nu }{2})}. \end{aligned}$$

• (iii) If pdf of$\fancyscript{L}(X_1)$is$f_{\nu ,a,\sigma ^2}$, $a<0$, then, as$n\rightarrow \infty$,

  $$\begin{aligned} \fancyscript{L}\left(\frac{2|a|}{\sigma ^2}M_n-\ln {n}-\left(\frac{\nu }{2}+1\right)\ln {\ln {n}}+\ln {K_3}\right)\Rightarrow \Lambda , \end{aligned}$$

  where$\Lambda$is the Gumbeldistribution

  $$\begin{aligned} \Lambda (x)=e^{-e^{-x}}, \quad x\in R^{1}, \end{aligned}$$

  and

  $$\begin{aligned} K_3=\frac{\nu ^{\frac{\nu }{2}}\sigma ^{\frac{\nu }{2}+3}}{2^{\nu +2}\Gamma (\frac{\nu }{2})}. \end{aligned}$$

Proof

• (i) From Proposition 2.3 (i) with $d=1$ and the l’Hospital’s rule we have, as $x\rightarrow \infty$,

  $$\begin{aligned} \int \limits ^{\infty }_{x}f_{\nu ,\sigma ^2}(u)\text{ d}u\sim \frac{c_{\nu ,\sigma }}{\nu \sigma }\left(\frac{x}{\sigma }\right)^{-\nu }=K_1x^{-\nu }, \end{aligned}$$

  (2.12)

  where

  $$\begin{aligned} c_{\nu ,\sigma }=\frac{\Gamma \left(\frac{\nu +1}{2}\right)}{\sqrt{\pi }\Gamma \left(\frac{\nu }{2}\right)\sigma }. \end{aligned}$$

  The statement (i) is standard for Pareto-like distributions (see, e.g., [1, 2]).

• (ii) From Corollary 2.4 (i) and the l’Hospital’s rule we have that, as $x\rightarrow \infty$,

  $$\begin{aligned} \int \limits ^{\infty }_{x}f_{\nu ,\sigma ^2,a}(u)\text{ d}u\sim K_2x^{-\frac{\nu }{2}} \end{aligned}$$

  (2.13)

  and the conclusion is analogs to (i).

• (iii) From Corollary 2.4 (iii) and the l’Hospital’s rule we find that, as $x\rightarrow \infty$,

  $$\begin{aligned} \int \limits ^{\infty }_{x}f_{\nu ,\sigma ^2,a}(u)\text{ d}u\sim \frac{\sigma }{2|a|\Gamma (\frac{\nu }{2})}\left(\frac{\nu |a|}{2\sigma }\right)^{\frac{\nu }{2}}x^{-\frac{\nu }{2}-1}\exp \left\{ -\frac{2|a|x}{\sigma ^2}\right\} . \end{aligned}$$

  (2.14)

  The statement (iii) is standard for gamma-like distributions (see, e.g., [1, 2]). $\square$

Now let us recall main results on limit theorems for record values in the sequences of i.i.d. random variables $\{X_n,n\ge 1\}$ with a common continuous distribution function $F$ which will be applied to the case of Student’s $t$-distributions.

The record times are $L_1=1$, $L_{n+1}=\min \left\{ k:k>n,X_k>X_{L_n}\right\}$ for $n=1,2,\ldots$, and the record values are $R_n=X_{L_n}$, $n=1,2,\ldots$. Let $W(x)=-\log (1-F(x))$ be the integrated hazard function and the associate distribution function $A(x)=1-e^{-\sqrt{W(x)}}$, $x\in R^1$. Let $l_{a,b}(x)=ax+b$, $a>0$, $b\in R^1$, be a group of affine homeomorphisms of $R^1$ with the composition law

$$\begin{aligned} l_{a_1,b_1}*l_{a_2,b_2}=l_{a_1a_2,a_1b_2+b_1}, \end{aligned}$$

the unit element $l_{1,0}$ and the inverse $l^{-1}_{a,b}=l_{a^{-1},a^{-1}b}$.

The domain of attraction problem for record values using linear normalization was solved by Resnick (see [3] also [4]). It was proved that the class of all possible non-degenerated weak limit laws $Q$ such that for suitable constants $a_n>0$, $b_n\in R^1$, as $n\rightarrow \infty$,

$$\begin{aligned} \fancyscript{L}\left(l^{-1}_{a_n,b_n}(R_n)\right)\Rightarrow Q \end{aligned}$$

coincide with the class of laws $\Phi \left(-\log (-\log {G(\cdot )})\right)$, where $\Phi$ is a standard normal distribution and $G$ is a $l$-max stable law, i. e. a non-degenerated distribution on $R^1$ such that for any $n\ge 2$ there exist constants $a_n>0$, $b_n\in R^1$ satisfying

$$\begin{aligned} G^n(x)=G\left(l_{a_n,b_n}(x)\right), \quad x\in R^1. \end{aligned}$$

As in the classical extreme value theory this class can be factorized into three linear types, saying that probability distributions $F_1$ and $F_2$ are of the same linear type it there exist constants $a>0$, $b\in R^1$ such that

$$\begin{aligned} F_1(x)=F_2\left(l_{a,b}(x)\right), \quad x\in R^1. \end{aligned}$$

In the classical case these types are generated by the Fréchet distribution $\Phi _\gamma$, the Gumbel distribution $\Lambda$ and the Weibull distribution

$$\begin{aligned} \Psi _{\gamma }(x)=\left\{ \begin{array}{l} 1, \quad \mathrm if \quad x\ge 0, \\ \exp \left\{ -(-x)^{\gamma }\right\} , \quad \mathrm if \quad x<0, \quad \gamma >0, \end{array} \right. \end{aligned}$$

which correspond to generators of three types of the limiting laws for $\fancyscript{L}\left(l_{a_n,b_n}(R_n)\right)$:

$$\begin{aligned} \tilde{\Phi }_{\gamma }(x)=\left\{ \begin{array}{l} 0,\quad \mathrm if \quad x\le 0, \\ \Phi (\log {x^{\gamma }}), \quad \mathrm if \quad x>0, \quad \gamma >0, \end{array} \right. \end{aligned}$$

$$\begin{aligned} \tilde{\Psi }_{\gamma }(x)=\left\{ \begin{array}{l} \Phi (\log (-x)^{\gamma }), \quad \mathrm if \quad x<0, \\ 1,\quad \mathrm if \quad x\ge 0, \quad \gamma >0, \end{array} \right. \end{aligned}$$

and the standard normal distribution $\Phi (x)$, $x\in R^1$.

We say that $F$ belongs to the record domain of attraction under linear normalization of the non-degenerated distribution $Q$ ($F\in \text{ RDA}_l(Q)$ for short) if there exist constants $a_n>0$ and $b_n\in R^1$ such that $\fancyscript{L}(l^{-1}_{a_n,b_n}(R_n))\Rightarrow Q$, as $n\rightarrow \infty$.

Duality theorem of Resnick says that $F\in \text{ RDA}_l(\tilde{\Phi }_{\gamma })\Leftrightarrow A\in \text{ MDA}_l(\Phi _{\frac{\gamma }{2}})$, $F\in \text{ RDA}_l(\tilde{\Psi }_{\gamma })\Leftrightarrow A \in \text{ MDA}_l(\Psi _{\frac{\gamma }{2}})$ and $F\in \text{ RDA}_l(\Phi )\Leftrightarrow A\in \text{ MDA}_l(\Lambda )$, where $\text{ MDA}_l(Q)$ denotes the maximum domain of attraction under linear normalization of the non-degenerated distribution $Q$ (see, e.g., [3]). As a corollary we find that in the case of heavy-tailed distributions $F$ the record values cannot have non-degenerate limiting distributions if we use linear normalization. Indeed, for the Pareto-like distributions $F$, satisfying, as $x\rightarrow \infty$,

$$\begin{aligned} 1-F(x)\sim Kx^{-\delta }, \quad \delta >0, \end{aligned}$$

the associate distributions $A$ satisfy, as $x\rightarrow \infty$,

$$\begin{aligned} 1-A(x)\sim e^{-\sqrt{\delta \log {x}}}. \end{aligned}$$

In this case $A\bar{\in }\text{ MDA}_l(\Phi _{\frac{\gamma }{2}})\cup \text{ MDA}_l(\Psi _{\frac{\gamma }{2}})\cup \text{ MDA}_l(\Lambda )$. This fact is an argument to consider limit theorems for the record values using power normalization.

Let

$$\begin{aligned} p_{\alpha ,\beta }(x)=\alpha |x|^{\beta }\mathrm sign {x}, \quad \alpha >0, \quad \beta >0, \quad x\in R^1. \end{aligned}$$

Observe that this class of functions form a group of homeomorphisms of $R^1$ with the composition law

$$\begin{aligned} p_{\alpha _1,\beta _1}*p_{\alpha _2,\beta _2}=p_{\alpha _1\alpha _2^{\beta _1},\beta _1\beta _2}, \end{aligned}$$

the unit element $p_{1,1}$ and the inverse

$$\begin{aligned} p_{\alpha ,\beta }^{-1}=p_{\alpha ^{-\beta ^{-1}},\beta ^{-1}}. \end{aligned}$$

We say that $F$ belongs to the record domain of attraction under power normalization of the non-degenerate distribution $Q$ ($F\in \text{ RDA}_p(Q)$ for short) if there exist constants $\alpha _n>0$, $\beta _n>0$ such that, as $n\rightarrow \infty$, $\fancyscript{L}\left(p^{-1}_{\alpha _n,\beta _n}(R_n)\right)\Rightarrow Q$.

A non-degenerate distribution function $\tilde{G}$ on $R^1$ is called $p$-max stable if for any $n\ge 2$ there exist constants $\tilde{\alpha }_n>0$, $\tilde{\beta }_n>0$ such that

$$\begin{aligned} \tilde{G}^n(x)=\tilde{G}(p_{\tilde{\alpha }_n,\tilde{\beta }_n}(x)), \quad x\in R^1. \end{aligned}$$

Probability distributions $F_1$ and $F_2$ are of the same power type if there exist constants $\alpha >0$, $\beta >0$ such that $F_1(x)=F_2(p_{\alpha ,\beta }(x))$, $x\in R^1$.

The class of non-degenerated limiting distributions for $\fancyscript{L}(p^{-1}_{\alpha _n,\beta _n}(R_n))$, as $n\rightarrow \infty$, is equal to the class of laws $\hat{\Phi} (-\log (-\log \tilde{G}(\cdot)))$, where $\tilde{G}$ is a $p$-max stable law K, and is factorized to the six power types, generated by the distribution functions (see [5, 6]):

$$\begin{aligned} \hat{\Phi }_{1,\gamma }(x)&=\left\{ \begin{array}{l} 0, \quad \mathrm if \quad x\le 1, \\ \Phi (\gamma \log {\log {x}}), \quad \mathrm if \quad x>1, \quad \gamma >0, \end{array} \right.\\ \hat{\Phi }_{2,\gamma }(x)&=\left\{ \begin{array}{l} 0, \quad \mathrm if \quad x\le 0,\\ \Phi (-\gamma \log |\log {x}|), \quad \mathrm if \quad 0<x<1,\\ 1, \quad \mathrm if \quad x\ge 1, \quad \gamma >0, \end{array} \right.\\ \hat{\Phi }_{3,\gamma }(x)&=\left\{ \begin{array}{l} 0, \quad \mathrm if \quad x\le -1,\\ \Phi (-\gamma \log {|\log {|x|}|}), \quad \mathrm if \quad -1<x<0,\\ 1, \quad \mathrm if \quad x\ge 0, \quad \gamma >0, \end{array}\right. \\ \hat{\Phi }_{4,\gamma }(x)&=\left\{ \begin{array}{l} \Phi (-\gamma \log {\log {|x|}}), \quad \mathrm if \quad x<-1,\\ 1, \quad \mathrm if \quad x\ge -1, \quad \gamma >0, \end{array} \right. \\ \hat{\Phi }_5(x)&=\left\{ \begin{array}{l} 0, \quad \mathrm if \quad x\le 0,\\ \Phi (\log {x}), \quad \mathrm if \quad x>0, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \hat{\Phi }_6(x)&=\left\{ \begin{array}{l} \Phi (-\log {|x|}), \quad \mathrm if \quad x<0,\\ 1, \quad \mathrm if \quad x\ge 0. \end{array} \right. \end{aligned}$$

There are the valid analog of Resnick’s duality theorem and the principle of equivalent tails, which says that if continuous distribution functions $F_1$ and $F_2$ are such that $r(F_1)=r(F_2)$ and $1-F_1(x)\sim 1-F_2(x)$, as $x\uparrow r(F_1)$, then $F_1\in \text{ RDA}_p(Q)$ if and only if $F_2\in \text{ RDA}_p(Q)$ with the same normalizing constants, where $r(F)=\sup \{x:F(x)<1\}$ and $Q$ is a non-degenerate limiting distribution for record values using power normalization.

The following analog of classical R. von Mises theorem [7] holds true.

Theorem 2.6

[8]. Assume that the integrated hazard function $W(x)$ is differentiable in some neighborhood of $r(F)$. Then:

• (i) if$r(F)=\infty$and

  $$\begin{aligned} \lim \limits _{x\rightarrow \infty }\frac{W^{\prime }(x)x\log {x}}{\sqrt{W(x)}}=\gamma , \quad \gamma >0, \end{aligned}$$

  then$F\in \text{ RDA}_p(\hat{\Phi }_{1,\gamma })$;

• (ii) if$0<r(F)<\infty$and

  $$\begin{aligned} \lim \limits _{x\uparrow r(F)}\frac{W^{\prime }(x)x\log \left(\frac{r(F)}{x}\right)}{\sqrt{W(x)}}=\gamma , \quad \gamma >0, \end{aligned}$$

  then$F\in \text{ RDA}_p(\hat{\Phi }_{2,\gamma })$;

• (iii) if$r(F)=0$and

  $$\begin{aligned} \lim \limits _{x\uparrow 0}\frac{W^{\prime }(x)x\log |x|}{\sqrt{W(x)}}=\gamma , \quad \gamma >0, \end{aligned}$$

  then$F\in \text{ RDA}_p(\hat{\Phi }_{3,\gamma })$;

• (iv) if$r(F)<0$ and

  $$\begin{aligned} \lim \limits _{x\uparrow r(F)}\frac{W^{\prime }(x)|x|\log \left(\frac{x}{r(F)}\right)}{\sqrt{W(x)}}=\gamma , \quad \gamma >0, \end{aligned}$$

  then$F\in \text{ RDA}_p(\hat{\Phi }_{4,\gamma })$;

• (v) if$W$is twice differentiable in some neighborhood of$r(F)$and

  $$\begin{aligned} \lim \limits _{x\uparrow r(F)}W(x)\left(\frac{W^{\prime \prime }(x)}{\left(W^{\prime }(x)\right)^2}+\frac{1}{xW^{\prime }(x)}\right)=0, \end{aligned}$$

  (2.15)

  then for$0<r(F)\le \infty$$F\in \text{ RDA}_p(\hat{\Phi }_5)$and for$r(F)\le 0$$F\in \text{ RDA}_p(\hat{\Phi }_6)$.

Proposition 2.7

• (i) If pdf of$F$is$f_{\nu ,\sigma ^2}$, then$F\in \text{ RDA}_p(\hat{\Phi }_5)$.

• (ii) If pdf of$F$is$f_{\nu ,\sigma ^2,a}$, $a>0$, then$F\in \text{ RDA}_p(\hat{\Phi }_5)$.

• (iii) If pdf of$F$is$f_{\nu ,\sigma ^2,a}$, $a<0$, then$F\in \text{ RDA}_l(\Phi )$.

Proof

• (i) From the principle of equivalent tails and (2.12) it is enough to check (2.15) with $r(F)=\infty$ and the integrated hazard function

  $$\begin{aligned} W(x)=\nu \ln {x}-\ln {K_1}. \end{aligned}$$

  Indeed,

  $$\begin{aligned} \frac{W^{\prime \prime }(x)}{\left(W^{\prime }(x)\right)^2}+\frac{1}{xW^{\prime }(x)}=\frac{-\frac{\nu }{x^2}}{\left(\frac{\nu }{x}\right)^2}+\frac{1}{\nu }\equiv 0. \end{aligned}$$

• (ii) From the principle of equivalent tails and (2.13) it is enough to check (2.15) with $r(F)=\infty$ and the integrated hazard function

  $$\begin{aligned} W(x)=\frac{\nu }{2}\ln {x}-\ln {K_2}. \end{aligned}$$

  Again we find that

  $$\begin{aligned} \frac{W^{\prime \prime }(x)}{\left(W^{\prime }(x)\right)^2}+\frac{1}{xW^{\prime }(x)}=\frac{-\frac{\nu }{2x^2}}{\left(\frac{\nu }{2x}\right)^2}+\frac{2}{\nu }\equiv 0. \end{aligned}$$

• (iii) From (2.14) and the principle of equivalent tails it is enough to consider the integrated hazard function

  $$\begin{aligned} W(x)=\left(\frac{\nu }{2}+1\right)\ln {x}+\frac{2|a|}{\sigma ^2}x-\ln {K_3}, \end{aligned}$$

  where

  $$\begin{aligned} K_3=\frac{\sigma }{2|a|\Gamma \left(\frac{\nu }{2}\right)}\left(\frac{\nu |a|}{2\sigma }\right)^{\frac{\nu }{2}}. \end{aligned}$$

  The corresponding associated distribution

  $$\begin{aligned} 1-A(x)&=\exp \left\{ -\sqrt{\left(\frac{\nu }{2}+1\right)\ln {x}+\frac{2|a|}{\sigma ^2}x-\ln {K_3}}\right\} \\&\sim \exp \left\{ -\sqrt{\frac{2|a|}{\sigma ^2}x}\right\} , \quad \mathrm as \quad x\rightarrow \infty . \end{aligned}$$

  Using again the principle of equivalent tails, Resnick’s duality theorem and criteria from the classical extreme value theory we easily find that $A\in \text{ MDA}_l(\Lambda )$ and thus $F\in \text{ RDA}_l(\Phi )$.$\square$