Research on the rate of convergence in the limit theorem for the maximum of random variables

Table of Contents

1. Introduction. Problem statement.
2. Used facts and notations.
3. Construction of asymptotic expansions and accompanying measures.
4. Weibull distribution.
5. log-Weibull distribution.
6. log-k-Weibull distribution.
7. Conclusion. Main results.

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1. Introduction. Problem statement.

In this paper, the rate of convergence in the Fisher-Tippett-Gnedenko limit theorem for the maxima of random variables is investigated. As is known from the theorem, maxima can converge to three different parametric types of distributions. The Gumbel limit distribution is of greatest interest, as maxima of a large and diverse class of distributions converge to it. One of the main problems in extreme value theory is the classification of all distributions converging to the Gumbel distribution. Since the domain of such distributions is extremely wide, the tasks of applications require dividing it into subdomains that are reasonable in a certain sense. An alternative to the division into subdomains can be scaling, that is, the introduction of indicators of the behavior of distribution tails. This paper discusses the issue of constructing such a scale.

The paper describes an approach using the von Mises structure to obtain an accompanying law that has a simple form and at the same time yields better approximations. Classes of distributions of Weibull and log-Weibull types are considered as examples, and a scale of classes of distributions from the Gumbel maximum domain of attraction continuing these two classes is also proposed. For each class, the rate of convergence to the Gumbel distribution and the rate of approach to the obtained accompanying law are calculated.

2. Used facts and notations.

Let $X_{1},...,X_{n},...$ be independent identically distributed random variables with the distribution function $F(x)$. Denote $M_{n}:=\max(X_{i},i=1,...,n)$. By virtue of the Gnedenko-Fisher-Tippett theorem, see [1], [2], as well as [3], [4], if there exist sequences of positive numbers $a_{n}$ and real numbers $b_{n}$ such that the limit of the probability $P(M_{n}\leq a_{n}x+b_{n})$ as $n\rightarrow\infty$ is a non-degenerate distribution function (takes more than two values), then this limit belongs to one of three types, namely, to the types of Fréchet, Weibull, and Gumbel distributions. In the present work, distributions belonging to the third type are considered, that is, those whose distribution functions $F$ satisfy for some sequences $a_{n}>0$ and $b_{n}$ and all $x$ the limit relation

$$\lim_{n\rightarrow\infty}P(M_{n}\leq a_{n}x+b_{n})=\Lambda(x):=\exp(-e^{-x}). \tag{1}$$

Such distributions are said to belong to the maximum domain of attraction of the Gumbel distribution, denoted as $F\in MDA(\Lambda)$. In addition, for the sake of simplicity of presentation, we will restrict ourselves to right-unbounded distributions, that is, $F(x)<1$ for all $x$. As can be seen from the following, the proposed approach can also be applied to distributions from the other two aforementioned maximum domains of attraction.

We consider the domain $MDA(\Lambda)$ since this domain is extremely wide, and the tasks of applications require its division into reasonable in a certain sense subdomains. Thus, this domain includes distributions with tails $1-F(x)$ roughly (logarithmically) equivalent as $x\rightarrow\infty$ to the tails of the Weibull distribution, that is, $\log(1-F(x))\sim-Cx^{p},$ $C,p>0,$ roughly equivalent to log-Weibull tails, that is, $\log(1-F(x))\sim-C(\log x)^{p},$ $C>0,p>1,$ while instead of powers on the right-hand sides one can take regularly varying at infinity functions, and a set of others, with even heavier (slower decreasing) and lighter (faster decreasing) tails at infinity.

As shown in [5], distributions from the domain $MDA(\Lambda)$ can be described using the von Mises representation: $F\in MDA(\Lambda)$ if and only if for some $x_{0}\geq0$ the following representation holds

$$1-F(x)=c(x)\exp\left\{ -\int_{x_{0}}^{x}\frac{1}{f(t)}dt\right\} ,\ \ x\geq x_{0}, \tag{2}$$

where $f(x)$ is a positive absolutely continuous function on $[x_{0},$ $\infty)$ such that $f^{\prime} (x)\rightarrow0,$ and $c(x)\rightarrow c>0$ as $x\rightarrow\infty$. Often it is more convenient to use a more flexible representation, namely, $F\in MDA(\Lambda)$ if and only if for some $x_{0}\geq0$,

$$1-F(x)=c(x)\exp\left\{ -\int_{x_{0}}^{x}\frac{g(t)}{f(t)}dt\right\} , \tag{3}$$

with the same properties of the functions $f(x)$ and $c(x),$ and $g(x)\rightarrow1$ as $x\rightarrow\infty.$ The normalizing sequences can be chosen as follows:

$$b_{n}=F^{\leftarrow}(1-n^{-1}),\ \ \ a_{n}=f(b_{n}), \tag{4}$$

see [4],[6]. Moreover, if relation (1) holds for the sequence $b_{n}$ indicated in (4) and some other sequence of positive $\tilde{a}_{n},$ then $a_{n}/\tilde{a}_{n}\rightarrow1$ as $n\rightarrow\infty,$ and, furthermore, there exists a von Mises representation (3) with other $\tilde{g}$ and $\tilde{f}$ such that $\tilde{a}_{n}=\tilde{f}(b_{n}).$

There is an extensive bibliography on the study of the quality of convergence in relation (1). In this paper, an approach is proposed in which, instead of studying the quality of approximation by the Gumbel distribution, better approximations are proposed. In the next section, an asymptotic expansion in the theorem on convergence to the Gumbel distribution is derived. Next, accompanying measures are proposed that yield a power-law rate of convergence.

3. Construction of asymptotic expansions and accompanying measures.

Suppose that representation (2) holds. Using expressions(4), we have that

$$\int_{x_{0}}^{b_{n}}\frac{1}{f(t)}dt=\log(nc(b_{n})). \tag{5}$$

Next,

$$G_{n}(x):=F^{n}(a_{n}x+b_{n})=\left( 1-c(a_{n}x+b_{n})\exp\left\{ -\int_{x_{0}}^{a_{n}x+b_{n}}\frac{1}{f(t)}dt\right\} \right) ^{n}.$$

Let us take the logarithm of this equality:

$$\log G_{n}(x)=n\log\left( 1-c(a_{n}x+b_{n})\exp\left\{ -\int_{x_{0}} ^{a_{n}x+b_{n}}\frac{1}{f(t)}dt\right\} \right) .$$

Denoting for brevity

$$g_{n}(x):=\int_{x_{0}}^{a_{n}x+b_{n}}\frac{1}{f(t)}dt - \log c(a_{n}x+b_{n}),$$

we have that

$$\log G_{n}(x)=n\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k+1}\left( -e^{-g_{n} (x)}\right) ^{k+1}=-ne^{-g_{n}(x)}\sum_{k=0}^{\infty}\frac{e^{-kg_{n}(x)} }{k+1}. \tag{6}$$

Using (5), we obtain:

$$\begin{aligned} g_{n}(x) & =\int_{x_{0}}^{b_{n}}\frac{1}{f(t)}dt +\int_{b_{n}}^{a_{n} x+b_{n}}\frac{1}{f(t)}dt-\log c(a_{n}x+b_{n})\\ & =\log n+\int_{b_{n}}^{a_{n}x+b_{n}}\frac{1}{f(t)}dt+\log c(b_{n})-\log c(a_{n}x+b_{n}). \end{aligned}$$

Let us introduce the function

$$\gamma_{n}(x)=\int_{b_{n}}^{a_{n}x+b_{n}}\frac{1}{f(t)}dt-\log\frac {c(a_{n}x+b_{n})}{c(b_{n})},$$

and rewrite (6) as follows:

$$\log G_{n}(x)=-e^{-\gamma_{n}(x)}\sum_{k=0}^{\infty}\frac{e^{-k\gamma_{n}(x)} }{(k+1)n^{k}}.$$

Since $a_{n}=f(b_{n}),$ we can write the expression for $\gamma_{n}(x)$ in the form:

$$\gamma_{n}(x)=\int_{b_{n}}^{a_{n}x+b_{n}}\left( \frac{1}{f(t)}-\frac {1}{f(b_{n})}\right) dt-\log\frac{c(a_{n}x+b_{n})}{c(b_{n})}+x. \tag{7}$$

Note that due to relation (1), for all $x$ we have that $\gamma _{n}(x)\rightarrow x$ as $n\rightarrow\infty.$ It is also easy to see that if we assume representation (3) instead of (2), then the expression for the normalizing constant $a_{n}$ changes to $a_{n}=f(b_{n})/g(b_{n})$, and the expression for $\gamma_{n}(x)$ — to the following:

$$\gamma_{n}(x)=\int_{b_{n}}^{a_{n}x+b_{n}}\left( \frac{g(t)}{f(t)} -\frac{g(b_{n})}{f(b_{n})}\right) dt-\log\frac{c(a_{n}x+b_{n})}{c(b_{n})}+x. \tag{8}$$

Thus,

$$\log G_{n}(x)=-e^{-\gamma_{n}(x)}\sum_{k=0}^{\infty}\frac{e^{-k\gamma_{n}(x)} }{(k+1)n^{k}};$$

and

$$G_{n}(x)=\exp\left( -e^{-\gamma_{n}(x)}\right) \exp\left( -\frac{1}{n} \sum_{k=0}^{\infty}\frac{e^{-(k+2)\gamma_{n}(x)}}{(k+2)n^{k}}\right) .$$

Let us now give another obvious expression for $\gamma_{n}(x)$ and, accordingly, for the accompanying measure.

Proposition 1. Let relation (1) be satisfied, then, if we set $a_{n}=f(b_{n}),$ then

$$\gamma_{n}(x)=-\log\frac{1-F(b_{n}+a_{n}x)}{1-F(b_{n})}=\log\frac {1}{n(1-F(b_{n}+a_{n}x))}. \tag{9}$$

Indeed, since

$$\gamma_{n}(x)=\int_{x_{0}}^{a_{n}x+b_{n}}\frac{g(t)dt}{f(t)}-\int_{x_{0} }^{b_{n}}\frac{g(t)dt}{f(t)}-\log c(a_{n}x+b_{n})+\log c(b_{n}),$$

then using representation (3), we obtain (9).

Thus, we obtain the accompanying law, which is given by the formula:

$$G_{n}(x) = \exp\left\{ -e^{-\gamma_{n}(x)}\right\}, \quad \gamma_{n}(x) =\log\frac{1}{n(1-F(b_{n}+a_{n}x))}. \tag{10}$$

As stated above, from(1) it follows that $\gamma_{n}(x) \to x, \quad n \to \infty$.

In the subsequent sections, classes of distributions are considered: Weibull, log-Weibull, and log-k-Weibull; for each type $\gamma_{n}(x)$ is calculated, and then the rate of convergence to the accompanying law $G_{n}(x)$ and the rate of convergence to the Gumbel distribution $\Lambda(x)$ are estimated.

4. Weibull distribution.

In this section, we will consider the Weibull distribution, which has the form:

$$1-F(x)=\mathbf{I}_{\{x\geq0\}}e^{-cx^{p}},\ \ \text{where }p>0,\ c>0$$

Hereafter we will omit the indicator everywhere, assuming that all equalities are valid for $\forall x \ge x_{0}$.

The von Mises representation will take the form:

$$1-F(x) = - \exp\left\{ -\int_{0}^{x}\frac{1}{Ct^{1-p}}dt \right\}, \quad C = \dfrac{1}{cp}$$

Let us find the sequences $b_{n}$ and $a_{n}$. To do this, we substitute $x = b_{n}$ into the distribution function and, taking into account that the following suits us

$$b_{n}=F^{\leftarrow}(1-n^{-1}), \ \quad \enspace a_{n} = f(b_{n}) = Cb_{n}^{1-p},$$

we obtain,

$$b_{n} = \bigg(\dfrac{1}{c} \log(n) \bigg)^{\frac{1}{p}}, \quad a_{n} = C \bigg(\dfrac{1}{c}\log(n) \bigg)^{\frac{1}{p} - 1}$$

Now let us calculate $\gamma_{n}(x)$ using expression (10):

$$\begin{aligned} \gamma_{n}(a_{n}x+b_{n} ) &= -\log(n) + c(a_{n}x+b_{n})^{p} = -\log(n) + c( Cb_{n}^{1-p}x + b_{n})^{p} = \\ &= -\log(n) + cb_{n}^p ( 1 + Cb_{n}^{-p}x)^{p} = -\log(n) + cb_{n}^{p}\bigg(1 + pCb_{n}^{-p}x + \dfrac{p(p-1)}{2}C^2b_{n}^{-2p}x^2 + O\bigg( \dfrac{1}{b_{n}^{3p}} \bigg) \bigg) = \\ &= x + C\dfrac{(p-1)}{2}b_{n}^{-p}x^2 + O\bigg( \dfrac{1}{b_{n}^{2p}} \bigg) = x + \dfrac{(p-1)}{2p}\dfrac{1}{\log(n)}x^2 + O\bigg( \dfrac{1}{(\log(n))^{2}} \bigg) \end{aligned}$$

Thus, we have obtained the expression for $\gamma_{n}(x)$ for the Weibull distribution. Next, we investigate the rate of convergence to the accompanying law $G_{n}(x) = \exp\left\{ -e^{-\gamma_{n}(x)}\right\}$ and to the limit law $\Lambda(x) = \exp\left\{- e^x\right\}$.

We have,

$$M_{n}(a_{n}x + b_{n}) = F^{n}(a_{n}x + b_{n}) = \exp\left\{ n\log(1-e^{-c(a_{n}x + b_{n})^{p}}) \right\}$$

Since $c(a_{n}x + b_{n})^{p} = \log(n) + \gamma_{n}(x)$, we obtain:

$$\begin{aligned} M_{n} &= \exp\left\{ n\bigg( -\dfrac{1}{n}e^{-\gamma_{n}(x)} - \dfrac{1}{2n^2}e^{-2\gamma_{n}(x)} + O\bigg(\dfrac{1}{n^3} \bigg) \bigg) \right\} = \\ &=\exp\left\{-e^{-\gamma_{n}(x)}\right\} \cdot \exp\left\{-\frac{e^{-2 \gamma_n(x)}}{2 n}+O\left(\frac{1}{n^{2}} \right)\right\} \end{aligned}$$

Proposition 2. The rate of approach to the accompanying law $G_{n}(x)=\exp\left\{ -e^{-\gamma_{n}(x)} \right\}$ for the Weibull distribution has the form:

$$|M_{n}(a_{n}x + b_{n}) - G_{n}(x) | = \exp\left\{ -e^{-\gamma_{n}(x)} \right\} \left| \dfrac{e^{-2\gamma_{n}(x)}}{2n}+ O\bigg(\dfrac{1}{n^2} \bigg) \right|$$

We have obtained the rate of convergence to the accompanying law of order $O\bigg(\dfrac{1}{n} \bigg)$.

Let us further estimate the rate of convergence to $\Lambda(x)$:

$$\begin{aligned} M_{n} &= \exp\left\{ -e^{-x}e^{ \frac{(p-1)}{2p}\frac{1}{\log(n)}x^2 + O\big( \frac{1}{(\log(n))^{2}} \big) }\right\} \exp\left\{ - \dfrac{1}{2n}e^{-2\gamma_{n}(x)} + O\bigg(\dfrac{1}{n^2} \bigg) \right\} = \\ &= \exp\left\{ -e^{-x} \right\} \exp\left\{ e^{-x}\frac{(p-1)}{2p}\frac{1}{\log(n)}x^2 + O\big( \frac{1}{(\log(n))^{2}} \big) \right\} \exp\left\{ - \dfrac{1}{2n}e^{-2\gamma_{n}(x)} + O\bigg(\dfrac{1}{n^2} \bigg) \right\} \end{aligned}$$

Proposition 3. The rate of convergence to the Gumbel distribution for the Weibull distribution has the form:

$$|M_{n}(a_{n}x + b_{n}) - \Lambda(x) | = \exp\left\{ -e^{-x} \right\} \left| \frac{(p-1)}{2p}\frac{1}{\log(n)}e^{-x}x^2 + O\bigg(\dfrac{1}{(\log(n))^2} \bigg) \right|$$

We obtained a logarithmic convergence rate, which is significantly slower than the power-law convergence rate to the accompanying law.

Note that only for $p=1$, the convergence rate can be better than logarithmic; it is easy to see that in this case $\gamma_n(x) = x$, then the convergence rate will be proportional to $n^{-1}$.

5. log-Weibull distribution.

Let us consider the log-Weibull distribution:

$$1-F(x)=\mathbf{I}_{\{x\geq0\}}e^{-c(\log(x))^{p}},\ \ \text{where }p > 1 ,\ c>0,$$

Similarly to the previous section, all equalities are valid for $\forall x \ge x_{0}$.

Let us write down the von Mises representation:

$$1-F(x) = - \exp\left\{ -\int_{0}^{x}\frac{1}{Ct \log(t)^{1-p}}dt \right\}, \quad C = \dfrac{1}{cp}$$

That is, the function $f(t) = Ct\log(t)^{1-p}$. Let us calculate $a_{n}$ and $b_{n}$:

$$b_{n} = \exp\left\{\bigg(\dfrac{1}{c} \log(n) \bigg)^{\frac{1}{p}} \right\}, \quad a_{n} = C\exp\left\{\bigg(\dfrac{1}{c} \log(n) \bigg)^{\frac{1}{p}} \right\} \bigg( \dfrac{1}{c} \log(n) \bigg)^{\frac{1}{p} -1}$$

We write out the expression for $\gamma_{n}(x)$:

$$\begin{aligned} \gamma_{n}(a_{n}x+b_{n} ) &= -\log n + c (\log(a_n x + b_{n}))^p = -\log n + c (\log b_n + \log(1 + C \log^{1-p} b_n x))^p = \\ &= -\log n + c \left(\log b_n + C x \log^{1-p} b_n - \frac{1}{2} C^2 x^2 \log^{2-2p} b_n + O(\log^{3-3p} b_n) \right)^p =\\ &= -\log n + c \log^{p} b_n \cdot \left(1+p C x \log^{-p} b_n - \frac{1}{2} p C^2 x^2 \log^{1-2p} b_n + O(\log^{2-3p} b_n) \right) = \\ &= -\log n + c \log^p b_n + x - \frac{1}{2} (c p)^{-1} x^2 \log^{1-p} b_n + O(\log^{2-2p} b_n) = \\ &= x - \dfrac{1}{2} (cp)^{-1} x^2 \log^{1-p} b_n + O(\log^{2-2p} b_n) \end{aligned}$$

Substituting the expression for $b_{n}$, we get:

$$\gamma_{n}(a_{n}x+b_{n} ) = x - \dfrac{1}{2} x^2 p^{-1} c^{-\frac{1}{p}} \log^{\frac{1}{p}-1} n + O(\log^{\frac{2}{p} - 2 } n)$$

Let us investigate the rate of convergence to the accompanying law and to the Gumbel distribution, respectively.

$$\begin{aligned} M_{n}(a_{n}x+b_{n}) &= F^{n}\left(a_{n} x+b_{n}\right)=\left(1-e^{-c\log(a_{n}x+b_{n})}\right)^{n}= \\ &=\left(1-e^{-\gamma_{n}(x)-\log n}\right)^{n}=e^{n \log \left(1-\frac{e^{-\gamma_{n}(x)}}{n}\right)}=e^{n\left(-\frac{e^{-\gamma_{n}(x)}}{n}-\frac{e^{-2 \gamma_{n}(x)}}{2 n^{2}}+O\left(\frac{1}{n^{3}}\right)\right)}= \\ &=e^{-e^{-\gamma_{n}(x)}-\frac{e^{-2 \gamma_{n}(x)}}{2 n}+O\left(\frac{1}{n^{2}}\right)}=\exp\left\{-e^{-\gamma_{n}(x)}\right\} \cdot \exp\left\{-\frac{e^{-2 \gamma_n(x)}}{2 n}+O\left(\frac{1}{n^{2}} \right)\right\} \end{aligned}$$

Proposition 4. The rate of approach to the accompanying law $G_{n}(x)=\exp\left\{ -e^{-\gamma_{n}(x)} \right\}$ for the log-Weibull distribution has the form:

$$|M_{n}(a_{n}x + b_{n}) - G_{n}(x) | = \exp\left\{ -e^{-\gamma_{n}(x)} \right\} \left| \dfrac{e^{-2\gamma_{n}(x)}}{2n}+ O\bigg(\dfrac{1}{n^2} \bigg) \right|$$

Proposition 5. The rate of convergence to the Gumbel distribution for the log-Weibull distribution has the form:

$$|M_{n}(a_{n}x + b_{n}) - \Lambda(x) | = \exp\left\{ -e^{-x} \right\} \left| \dfrac{1}{2} x^2 p^{-1} c^{-\frac{1}{p}} \log^{\frac{1}{p}-1} n + O(\log^{\frac{2}{p} - 2 } n) \right|$$

The obtained rate of convergence is of the order of $\log^{\frac{1}{p}-1} n$, which is significantly slower than the rate of convergence to the accompanying law $\frac{1}{n}$.

6. log-k-Weibull distribution.

One of the continuations of the scale started by the two classes of distributions considered above can be the following. In distributions with Weibull or log-Weibull type tails, the function $f(t)$ in the von Mises representation is equal to $C t^{1-p}, p>0$ and $C t \log ^{1-p} t, p>1$, respectively. Heavier tails of distributions from $MDA(\Lambda)$ are, for example, tails equivalent to $\exp \left(-C \log x \log_{(k)} ^{p} x\right),p>1$ for which $f(t)=C t\left(\log _{(k)} t\right)^{-p}$, where $\log _{k}=\overbrace{\log \ldots \log }^{k}$. Obviously, for any natural $k$, the function $f(t)=C t\left(\log _{(k)} t\right)^{-p}$ satisfies the conditions required in representations(2), (3). The number $k$ of logarithm repetitions in these expressions for $f(t)$ can be called the Gumbel index, then distributions with Weibull-type tails have a Gumbel index $k = 0$, log-Weibull — $k = 1$, and so on.

$$c(x) \equiv 1, \quad C = \frac{1}{c}, \quad f(t)=\frac{1}{c} t \log _{k}^{-p} t, \quad \log _{k} x_{0}=0, \quad k \geq 2$$

Proposition 6. Let us show that for $f(t)=\frac{1}{c} t \log _{k}^{-p} t$ the tails of the distribution $F(x)$ will have the representation:

$$\bar{F}(x)=\exp \left(-c \log x \log _{k}^{p} x\left(1+O\left(\log _{k}^{-1} x\right)\right)\right) \quad \text { as } x \rightarrow \infty, k=2,3, \dots$$

Substituting $f(t)$ into the von Mises representation and integrating by parts, we obtain:

$$\int_{x_{0}}^{x} \log _{k}^{p} t \frac{d t}{t}=\log x \log _{k}^{p} x-p \int_{x_{0}}^{x} \log _{k}^{p-1} t \frac{d \log t}{\log _{2} t \ldots \log _{k-1} t}$$

since $\log _{k-1} t \geq \log _{k-1} x_{0}=1,$ then $\log _{i} t>1, i=2, \ldots, k-2$, in which case the last integral can be estimated by the integral

$$\int_{x_{0}}^{x} \log _{k}^{p-1} t d \log t$$

we integrate by parts once more and obtain that the integral above can be estimated as $O\left(\log x \log _{k}^{p-1} x\right)$.

To find the rate of convergence with the accompanying law, let us find the expression for $\gamma_{n}$. We use formula (9):

$$\gamma_{n}(x)=\log \frac{1}{n\left(1-F\left(b_{n}+a_{n} x\right)\right)}=-\log (n)+c \log \left(a_{n} x+b_{n}\right) \log _{k}^{p}\left(a_{n} x+b_{n}\right)$$

Substituting $x = b_{n}$ into (2) and using expression(4), we write the formula for finding $b_{n}$.

$$\log (n)=c \log \left(b_{n}\right) \log _{(k)}^{p}\left(b_{n}\right)$$

For the log-k-Weibull distribution, it is not possible to find $b_{n}$ and $a_{n}$ from the above equation in explicit form, unlike the cases of the Weibull and log-Weibull distributions.

Let us write the expression for $\gamma_n(x)$ in terms of $b_{n}$. We have:

$$a_{n} = f(b_{n})=\frac{1}{c} b_{n} \log _{(k)}^{-p} b_{n}$$

Let us expand into a Taylor series $\log \left(a_{n} x+b_{n}\right)$:

$$\begin{aligned} \log \left(a_{n} x+b_{n}\right) &= \log b_{n}+\log \left(1+\frac{a_{n} x}{b_{n}}\right)=\log b_{n}+\log \left(1+ \frac{x}{c} \cdot \log_{(k)}^{-p} b_{n}\right) = \\ &= \log b_{n} + \frac{x}{c} \cdot \log_{(k)}^{-p} b_{n} - \frac{x^{2}}{2c^{2}} \cdot \log_{(k)}^{-2p} b_{n} + O(\log_{(k)}^{-3p} b_{n}) \end{aligned}$$

For $\log_{(k)} \left(a_{n} x+b_{n}\right)$ the expansion will have the form:

$$\begin{aligned} \log_{(k)} \left(a_{n} x+b_{n}\right) &= \log _{k-1}\left(\log \left(a_{n} x+b_{n}\right)\right)= \\ &=\log_{k-2}\left(\log \left( \log b_{n} + \frac{x}{c} \cdot \log_{(k)}^{-p} b_{n} - \frac{x^{2}}{2c^{2}} \cdot \log_{(k)}^{-2p} b_{n} + O(\log_{(k)}^{-3p} b_{n}) \right)\right) = \\ &= \log_{k-2}\left(\log_{(2)} b_{n} + \frac{x}{c} \cdot \log_{(k)}^{-p} b_{n} \cdot \log^{-1}{b_{n}} - \frac{x^2}{2c^2} \cdot \log_{(k)}^{-2p} b_{n} \cdot \log^{-1} b_{n} + O(\log_{(k)}^{-3p} b_{n} \cdot \log^{-1} b_{n} ) \right) = \\ &= \log_{(k)} b_{n} + \frac{x}{c} \cdot \log_{(k)}^{-p} b_{n}\cdot \sigma_{k-1}^{-1}{(b_{n})} - \frac{x^{2}}{2c^{2}} \cdot \log_{(k)}^{-2p} b_{n} \cdot \sigma_{k-1}^{-1}{(b_{n})} + O(\log_{(k)}^{-3p} b_{n} \cdot \sigma_{k-1}^{-1}{(b_{n})}) \end{aligned}$$

Where the following notations were used:

$$\sigma_{k}(x)=\prod_{i=1}^{k} \log _{(i)}(x), \quad \sigma_{1}(x)=\log (x)$$

Further,

$$\begin{aligned} \log _{(k)}^{p}\left(a_{n} x+b_{n}\right) &= \log_{(k)}^p b_{n} \cdot \left(1 + \frac{p x}{c} \cdot \log_{(k)}^{-p-1} b_{n} \cdot \sigma_{k-1}^{-1}{(b_{n})} - \right. \\ &\quad \left. - \frac{p x^2}{2c^2} \cdot \log_{(k)}^{-2p-1} b_{n} \cdot \sigma_{k-1}^{-1}{(b_{n})} + O(\log_{(k)}^{-3p-1} b_{n} \cdot \sigma_{k-1}^{-1}{(b_{n})}) \right) = \\ &= \log_{(k)}^p b_{n} + \frac{p x}{c} \cdot \underset{\sigma_{k}^{-1}{(b_{n})}}{\underbrace{ \log_{(k)}^{-1} b_{n} \cdot \sigma_{k-1}^{-1}{(b_{n})}}} - \frac{p x^2}{2c^2} \cdot \log_{(k)}^{-p} b_{n} \cdot \sigma_{k}^{-1}{(b_{n})} + O(\log_{(k)}^{-2p} b_{n} \cdot \sigma_{k}^{-1}{(b_{n})}) \end{aligned}$$

Now we can obtain the final representation for $\gamma_{n}$:

$$\begin{aligned} \gamma_{n} &= - \log n + c\left( \log b_{n} + \frac{x}{c} \cdot \log_{(k)}^{-p} b_{n} - \frac{x^{2}}{2c^{2}} \cdot \log_{(k)}^{-2p} b_{n} + O(\log_{(k)}^{-3p} b_{n}) \right) \times \\ &\quad \times \left( \log_{(k)}^p b_{n} + \frac{p x}{c} \cdot \log_{(k)}^{-1} b_{n} \cdot \sigma_{k}^{-1}{(b_{n})} - \frac{p x^2}{2c^2} \cdot \log_{(k)}^{-p} b_{n} \cdot \sigma_{k}^{-1}{(b_{n})} + O(\log_{(k)}^{-2p} b_{n} \cdot \sigma_{k}^{-1}{(b_{n})}) \right) = \\ &= - \log n + c \log b_{n} \log_{(k)}^p b_{n} + x - \frac{x^2}{2c} \cdot \log_{(k)}^{-p} b_{n} + O(\log_{(k)}^{-2p} b_{n}) = \\ &= x - \frac{x^2}{2c \log_{(k)}^{p} b_{n}} + O(\log_{(k)}^{-2p} b_{n}) \end{aligned}$$

Let us now consider the rate of convergence to $\Lambda(x)$, as well as to the obtained accompanying law $G_n(x)$.

For the distribution of maxima, we have the following representation:

$$M_{n}\left(a_{n} x+b_{n}\right)=F^{n}\left(a_{n} x+b_{n}\right)=\left(1-\exp \left\{-c \log \left(a_{n} x+b_{n}\right) \log _{k}^{p}\left(a_{n} x+b_{n}\right)\right\}\right)^{n}$$

Using that $\gamma_{n}(x)=-\log (n)+c \log (a_{n}x+b_n) \log _{k}^{p}(a_{n}x+b_n)$, we obtain:

$$\begin{aligned} F^{n}\left(a_{n} x+b_{n}\right) &= \left(1-e^{-\gamma_{n}(x)-\log b_{n} \log_{(k)}^{p} b_{n}}\right)^{n}= \\ &=\left(1-e^{-\gamma_{n}(x)-\log n}\right)^{n}=e^{n \log \left(1-\frac{e^{-\gamma_{n}(x)}}{n}\right)}=e^{n\left(-\frac{e^{-\gamma_{n}(x)}}{n}-\frac{e^{-2 \gamma_{n}(x)}}{2 n^{2}}+O\left(\frac{1}{n^{3}}\right)\right)}= \\ &=e^{-e^{-\gamma_{n}(x)}-\frac{e^{-2 \gamma_{n}(x)}}{2 n}+O\left(\frac{1}{n^{2}}\right)}=\exp\left\{-e^{-\gamma_{n}(x)}\right\} \cdot \exp\left\{-\frac{e^{-2 \gamma_n(x)}}{2 n}+O\left(\frac{1}{n^{2}} \right)\right\} \end{aligned}$$

Now let us consider the rate of approach to the accompanying law

$$\begin{aligned} F^{n}\left(a_{n} x+b_{n}\right)-e^{-e^{-\gamma_n(x)}} &= e^{-e^{-\gamma_{n}(x)}} \cdot\left(e^{-\frac{e^{-2 \gamma_{n}(x)}}{2 n}}-1\right)+O\left(\frac{1}{n^{2}}\right)= \\ &=e^{-e^{-\gamma_{n}(x)}} \cdot\left(1-\frac{e^{-2 \gamma_{n}(x)}}{2 n}+O\left(\frac{1}{n^{2}}\right)-1\right)+O\left(\frac{1}{n^{2}}\right) ; \end{aligned}$$

Proposition 7. For the log-k-Weibull distribution, the rate of approach to the accompanying law $G_{n}(x)$ is equal to

$$|M_{n}(a_{n}x + b_{n}) - G_{n}(x) | = \exp\left\{ -e^{-\gamma_{n}(x)} \right\} \left| \dfrac{e^{-2\gamma_{n}(x)}}{2n}+ O\bigg(\dfrac{1}{n^2} \bigg) \right|$$

Remark 1. Thus, the rate of convergence of $M_n$ to $G_n$ is of order $O\left(\frac{1}{n} \right)$

Now let us find the asymptotics of the rate of convergence of $M_n$ to the Gumbel distribution function.

From Remark 2, we have that

$$F^{n}\left(a_{n} x+b_{n}\right)=e^{-e^{-\gamma_{n}(x)}}+O\left(\frac{1}{n}\right), n \rightarrow \infty$$

Now let us find the asymptotics of the rate of convergence of $M_{n}$ to the Gumbel distribution function

$$\begin{aligned} F^{n}\left(a_{n} x+b_{n}\right)-\Lambda(x) &= e^{-e^{-\gamma_{n}(x)}}+O\left(\frac{1}{n}\right)-e^{-e^{-x}} = \\ &= e^{-e^{-x + \frac{x^2}{2c \log_{(k)}^{p} b_{n}} + O(\log_{(k)}^{-2p} b_{n})}}-e^{-e^{-x}} =\\ &= e^{-e^{-x}} \cdot \left(1 + \frac{x^2}{2c \log_{(k)}^{p} b_{n}} + O(\log_{(k)}^{-2p} b_{n}) \right) -e^{-e^{-x}} = \frac{x^2 e^{-e^{-x}}}{2c \log_{(k)}^{p} b_{n}} + O(\log_{(k)}^{-2p} b_{n}) \end{aligned}$$

Thus, the rate of convergence to the Gumbel distribution is of order $O(\log_{(k)}^{-p} b_{n})$, which is significantly slower than the rate of convergence of the maxima to the accompanying law $O\left(\frac{1}{n}\right)$

Remark 2. It is easy to show that $O\left(\frac{1}{\log _{k}^{p}\left(b_{n}\right)}\right)=O\left(\frac{1}{\log _{k}^{p}(n)}\right)$

This obviously follows from the equation for $b_{n}$:

$$\log (n)=c \log \left(b_{n}\right) \log _{(k)}^{p}\left(b_{n}\right)$$

Applying the logarithm to both sides, we get:

$$\log _{2}(n)=\log (c)+\log _{2}\left(b_{n}\right)+p \log _{(k+1)}\left(b_{n}\right)$$

From which it follows that $\log _{(k)}\left(b_{n}\right)=\log _{(k)}(n)+o(1)$

Proposition 8. For the log-k-Weibull distribution, the rate of convergence to the Gumbel distribution $\Lambda(x)$ for $k>1$ is equal to

$$\left|M_{n}\left(a_{n} x+b_{n}\right)-\Lambda(x)\right| = \exp\left\{ -e^{-x} \right\} \left| \frac{x^2}{2c \log_{(k)}^{p}{n}} + O(\log_{(k)}^{-2p}{n}) \right|$$

7. Conclusion. Main results.

In this paper, a sequence of accompanying laws in the B. V. Gnedenko limit theorem for the maxima of independent random variables is described, and the rate of convergence of the maxima to the accompanying law $G_{n}(x) = \exp\left\{ -e^{-\gamma_{n}(x)}\right\}$ for classes of distributions from the proposed scale was investigated. For all classes, the rate of convergence to the accompanying law is of the order of $\sim \dfrac{1}{n}$. The rates of convergence of the maxima to the Gumbel limit distribution were also calculated. These rates of convergence turned out to be significantly slower relative to the accompanying law. Thus, for the Weibull distribution, a rate of the order of $\sim \dfrac{1}{\log(n)}, p>1$ and $\sim n^{-1}, p=1$ was obtained, for the log-Weibull $\sim \log^{\frac{1}{p}-1}(n), p>1$, and for the log-k-Weibull $\sim \log_{(k)}^{-p}(n) , p>0$. Thus, the constructed accompanying law has a good rate of approach to the initial sequence of distributions, but at the same time has a simpler form, which is convenient to use for estimating the maxima of the considered types of distributions.

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References

1. Gnedenko B.V. Limit theorems for the maximal term of a variational series, DAN, 1941, vol. 32, No. 1, pp. 7 — 9.
2. Gnedenko B. V. Sur la distribution limite du terme maximum d’une serie aléatoire. Annals of Mathematics, 1943, vol. 44, No. 3, pp. 423 — 453.
3. L. de Haan, A. Ferreira. Extreme Value Theory. An Introduction. Springer, 2006, 418 p.
4. P. Embrechts, C. Klüppelberg, T. Mikosch. Modelling Extremal Events: for Insurance and Finance. Springer 2012. 641 p.
5. Balkema A. A., de Haan L. On R. von Mises’ condition for the domain of attraction of $\exp\{-e^{-x}\}.$ Ann. Math. Statist., 1972, vol. 43, pp. 1352 — 1354.
6. Resnick Sidney I. Extreme values, regular variation, and point processes. New-York, Berlin, Heidelberg: Springer-Verlag, 1987, 320 p.
7. Gnedenko, B. V. and Kolmogorov A. N. (1954) Limit Theorems for Sums of Independent Random Variables. Addison-Wesley, Cambridge, Mass.
8. W. J. Hall, Jon A. Wellner (1979) The rate of convergence in law of the maximum of an exponential sample. Statistica Neerlandica, 33, 3, 151—154.
9. Piterbarg V. I. (2015) Twenty Lectures About Gaussian Processes. Atlantic Financial Press London, New York, 2015.