# Limiting Distributions for the Minimum-Maximum Models

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Extreme Value Distribution for i.i.d. Sequences

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Extreme Value Distribution for Stationary Sequences

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Rate of Convergence of the Minimum-Maximum Model

**Theorem**

**4.**

**Proof.**

**Remark**

**2.**

**Corollary**

**1.**

**Proof.**

## 5. Numerical Experiments

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The sketchs above show the figures of ${G}_{A}^{\prime}(x)$ (red star) and ${f}_{n}(x)$ (blue line) for i.i.d. uniform sequence when $A=\frac{1}{2}$, 1 and 20, respectively.

**Figure 2.**The sketches above show the figures of ${G}_{A}^{\prime}(x)$ (red star) and ${f}_{n}(x)$ (blue line) for i.i.d. exponential sequence when $A=\frac{1}{2}$, 1 and 20, respectively.

**Figure 3.**The sketches above show the figures of ${G}_{A}^{\prime}(x)$ (red star) and ${f}_{n}(x)$ (blue line) for i.i.d. Cauchy’s sequence when $A=\frac{1}{2}$, 1 and 20, respectively.

Distribution Function F | Normalized Constant ${\mathit{a}}_{\mathit{n}}$ and ${\mathit{b}}_{\mathit{n}}$ |
---|---|

$F(x)=\left\{\begin{array}{cc}x,\phantom{\rule{1.em}{0ex}}\hfill & x\in [0,1]\hfill \\ 0,\hfill & \mathrm{others}\hfill \end{array}\right.$ | ${a}_{n}={n}^{-1-\frac{1}{An}}$, ${b}_{n}=1-{n}^{-\frac{1}{An}}$ |

$F(x)=\left\{\begin{array}{cc}1-{e}^{-x},\hfill & \phantom{\rule{3.33333pt}{0ex}}x\ge 0\hfill \\ 0,\hfill & \phantom{\rule{3.33333pt}{0ex}}x<0\hfill \end{array}\right.$ | $a}_{n}=\frac{1}{n$, ${b}_{n}=\frac{1}{An}lnn$ |

$F(x)=\frac{1}{\pi}arctanx+\frac{1}{2},x\in R$ | $\phantom{\rule{2.em}{0ex}}{a}_{n}={n}^{-1-\frac{1}{An}}\pi (1+{cot}^{2}({n}^{-\frac{1}{An}}\pi )),\phantom{\rule{3.33333pt}{0ex}}{b}_{n}=cot({n}^{-\frac{1}{An}}\pi )$ |

$\mathit{A}=1/2$ | $\mathit{A}=1$ | $\mathit{A}=20$ | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | Rate | Order | $\mathit{n}$ | Rate | Order | $\mathit{n}$ | Rate | Order |

2 | 0.3078 | 1 | 0.3679 | 1 | 0.3679 | |||

10 | 0.00516 | $-1.1097$ | 10 | 0.0246 | $-1.1748$ | 10 | 0.0192 | $-1.2824$ |

${10}^{2}$ | 0.0048 | $-1.0314$ | ${10}^{2}$ | 0.0024 | $-1.0107$ | ${10}^{2}$ | 0.0018 | $-1.0280$ |

${10}^{3}$ | $4.7561\times {10}^{-4}$ | $-1.0040$ | ${10}^{3}$ | $2.3545\times {10}^{-4}$ | $-1.0083$ | ${10}^{3}$ | $1.8402\times {10}^{-4}$ | $-0.9904$ |

${10}^{4}$ | $4.7526\times {10}^{-5}$ | $-1.0003$ | ${10}^{4}$ | $2.3535\times {10}^{-5}$ | $-1.0002$ | ${10}^{4}$ | $1.8395\times {10}^{-5}$ | $-1.0002$ |

$\mathit{A}=1/2$ | $\mathit{A}=1$ | $\mathit{A}=20$ | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{n}$ | Rate | Order | $\mathit{n}$ | Rate | Order | $\mathit{n}$ | Rate | Order |

2 | 0.1617 | 1 | 0.3679 | 1 | 0.3679 | |||

10 | 0.0280 | $-1.0896$ | 10 | 0.0279 | $-1.1195$ | 10 | 0.0248 | $-1.1718$ |

${10}^{2}$ | 0.0027 | $-1.0136$ | ${10}^{2}$ | 0.0027 | $-1.0107$ | ${10}^{2}$ | 0.0027 | $-1.0280$ |

${10}^{3}$ | $2.7070\times {10}^{-4}$ | $-1.0013$ | ${10}^{3}$ | $2.7070\times {10}^{-4}$ | $-1.0083$ | ${10}^{3}$ | $2.5762\times {10}^{-4}$ | $-0.9904$ |

${10}^{4}$ | $2.7060\times {10}^{-5}$ | $-1.0002$ | ${10}^{4}$ | $2.7060\times {10}^{-5}$ | $-1.0002$ | ${10}^{4}$ | $1.9649\times {10}^{-5}$ | $-1.0002$ |

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Peng, L.; Gao, L. Limiting Distributions for the Minimum-Maximum Models. *Mathematics* **2019**, *7*, 719.
https://doi.org/10.3390/math7080719

**AMA Style**

Peng L, Gao L. Limiting Distributions for the Minimum-Maximum Models. *Mathematics*. 2019; 7(8):719.
https://doi.org/10.3390/math7080719

**Chicago/Turabian Style**

Peng, Ling, and Lei Gao. 2019. "Limiting Distributions for the Minimum-Maximum Models" *Mathematics* 7, no. 8: 719.
https://doi.org/10.3390/math7080719