Sentence examples for using the exponential inequality from inspiring English sources

Then, using the exponential inequality, we obtain the Rosenthal inequality (Proposition 2.2).

By using the exponential inequality, we investigate the Berry-Esséen bound of sample quantiles for negatively associated (NA) random variables and obtain the rate O ( n − 1 / 6 log n ).

Furthermore, we will study the complete convergence for acceptable random variables using the exponential inequalities.

Using the exponential inequalities, we further study the complete convergence for acceptable random variables.

By using the exponential inequalities, they extended Theorem 1.1 for independent random variables to NOD random variables without condition (1.3).

In Section 3, we will study the complete convergence for acceptable random variables using the exponential inequalities established in Section 2. In this section, we will present some exponential inequalities for acceptable random variables, such as Bernstein-type inequality and Hoeffding-type inequality.

Furthermore, for every integer n ≥ t 0, using the exponential martingale inequality (see [3], Theorem 1.7.4) on sees that P { sup t 0 ≤ t ≤ n [ M ( t ) − 2 ∫ t 0 t | x T ( s ) g ( x ( s ), s ) | 2 ( 1 + | x ( s ) | 2 ) 2 d s ] > 2 log n } ≤ 1 n 2.

By using the exponential mean-square stability definition, the stability condition with an H∞ performance is guaranteed for the fuzzy system with the sampled-data fuzzy filter, and its sufficient condition is converted into the linear matrix inequality (LMI) format.

The exponential function is available in bipolar technology using the exponential characteristic of the bipolar transistor.

We have obtained general versions of the von Bahr-Esseen moment inequality, the exponential inequality, and convergence theorems.

Moreover, by constructing one-parameter families of operator monotone functions, we will get many operator inequalities; especially, we will extend the Furuta inequality and the exponential inequality of Ando.