An Upper Bound Estimation About the Sample Average of Interval-Valued Random Sets

Abstract

In this paper, we give an upper bound estimation about the probability of the event that the sample average of i.i.d. interval-valued random sets is included in a closed set. The main tool is Cramér theorem in the classic theory of large deviation principle about real-valued random variables.

Appendix

Cramér theorem: Let $X_1, X_2,\cdots ,$ be i.i.d random variables and satisfy $Ee^{\lambda |X_1|}<\infty$ for some $\lambda >0.$ Then for any closed set $F\subset \mathbb {R}$, we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log P\{\frac{1}{n}\sum _{i=1}^nX_i\in F\}\le -\inf \limits _{x\in F} I(x), \end{aligned}$$

and for any open set $G\subset \mathbb {R}$, we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }\frac{1}{n}\log P\{\frac{1}{n}\sum _{i=1}^nX_i\in G\}\ge -\inf \limits _{x\in G} I(x), \end{aligned}$$

where

$$\begin{aligned} I(x)=\sup _{\lambda \in \mathbb {R}}\{\lambda x-\log Ee^{\lambda X_1}\}. \end{aligned}$$