# On Max-Semistable Laws and Extremes for Dynamical Systems

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## Abstract

**:**

## 1. Introduction

#### 1.1. Overview on the Theory of Extremes

#### 1.2. Max-Semistable Laws and Corresponding Evt

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 2. Convergence to a Max-Semistable Law for Dynamical Systems

#### 2.1. Main Results

**Theorem**

**1.**

**Corollary**

**1.**

#### 2.2. Proof of Theorem 1

- An error term ${\mathcal{E}}_{1}$ which depends on the decay of correlations associated to separating the blocks by lag t. This is bounded by$${\mathcal{E}}_{1}\le C(p,q)\parallel {\phi}_{1}{\parallel}_{BV}{\parallel {\phi}_{2}\parallel}_{{L}^{1}}{\tau}_{1}^{-n}$$
- An error term ${\mathcal{E}}_{2}$ associated to the decomposition in (9). This is bounded as follows$${\mathcal{E}}_{2}\le n\sum _{j=2}^{p}\mu ({X}_{1}>{u}_{n},{X}_{j}>{u}_{n}).$$For observables of the form $\varphi \left(x\right)=\psi \left(\mathrm{dist}(x,\tilde{x})\right)$, it is shown that for $\mu $-a.e. $\tilde{x}\in X$ that ${\mathcal{E}}_{2}=O\left({n}^{-{\gamma}_{1}}\right)$ for some ${\gamma}_{1}>0$. See [18].
- A remainder error term of the form $max\{p,qt\}\mu ({X}_{1}>{u}_{n})$ which arises from the requirement that $p,q,t$ are integers. By choice of $p,q,t$ and ${u}_{n}$, we see that ${\mathcal{E}}_{3}=O\left({n}^{-{\gamma}_{2}}\right)$ for some ${\gamma}_{2}>0$.

#### 2.3. Proof of Corollary 1

**Lemma**

**1.**

#### 2.4. On the Role of the Extremal Index

**Definition**

**1.**

## 3. Numerical Studies

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Numerical estimates of the tail index ξ (

**A**) and the extremal index θ (

**B**) for the process $\left({X}_{i}\right)$ defined in Example 4. Grey bands mark the 95% confidence intervals around the obtained estimates.

**Figure 2.**Numerical estimates of the tail index ξ (

**A**) and the extremal index θ (

**B**) for the process for the process $\left({Y}_{i}\right)$ defined in Example 5.

**Figure 3.**As Figure 1, but for the process $\left({X}_{i}\right)$ defined in Example 6 with $\tilde{x}={\textstyle \frac{1}{2}}\sqrt{3}$ (

**A**,

**B**) and $\tilde{x}={\textstyle \frac{1}{2}}$ (

**C**,

**D**).

**Figure 4.**As Figure 1, but for the process $\left({X}_{i}\right)$ defined in Example 7 with $\tilde{x}={\textstyle \frac{1}{2}}\sqrt{3}$ (

**A**,

**B**) and $\tilde{x}={\textstyle \frac{3}{4}}$ (

**C**,

**D**).

**Figure 5.**As Figure 1, but for the process $\left({X}_{i}\right)$ defined in Example 8 with $\tilde{x}={\textstyle \frac{1}{2}}\sqrt{3}$ (

**A**,

**B**) and $\tilde{x}=3-\sqrt{8}$ (

**C**,

**D**).

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Holland, M.P.; Sterk, A.E.
On Max-Semistable Laws and Extremes for Dynamical Systems. *Entropy* **2021**, *23*, 1192.
https://doi.org/10.3390/e23091192

**AMA Style**

Holland MP, Sterk AE.
On Max-Semistable Laws and Extremes for Dynamical Systems. *Entropy*. 2021; 23(9):1192.
https://doi.org/10.3390/e23091192

**Chicago/Turabian Style**

Holland, Mark P., and Alef E. Sterk.
2021. "On Max-Semistable Laws and Extremes for Dynamical Systems" *Entropy* 23, no. 9: 1192.
https://doi.org/10.3390/e23091192