- Open Access
Convergence rate of extremes from Maxwell sample
© Liu and Liu; licensee Springer. 2013
Received: 30 May 2013
Accepted: 1 October 2013
Published: 7 November 2013
For the partial maximum from a sequence of independent and identically distributed random variables with Maxwell distribution, we establish the uniform convergence rate of its distribution to the extreme value distribution.
MSC:62E20, 60E05, 60F15, 60G15.
One interesting problem in extreme value theory is to consider the convergence rate of some extremes. For the uniform convergence rate of extremes under the second-order regular variation conditions, see Falk , Balkema and de Haan , de Haan and Resnick  and Cheng and Jiang . For the extreme value distributions and their associated uniform convergence rates for given distributions, see Hall and Wellner , Hall , Peng et al. , Lin and Peng  and Lin et al. .
The MD and the convergence rate of extremes from Maxwell sample have been widely used in the field of physics. We establish the uniform convergence rate of its distribution to the extreme value distribution and give an improved proof for the pointwise convergence rate of MD.
This paper is organized as follows. Section 2 gives some auxiliary results. In Section 3, we present the main result. Related proofs are given in Section 4.
2 Auxiliary results
To establish the uniform convergence of to its extreme value distribution , we need some auxiliary results. The first result is the decomposition of , which is the following result.
For simplicity, throughout this paper, let C be a generic positive constant whose value may change from line to line, and let , (, ) be absolute positive constants.
where , , . So, , , implying . For large n, we have the following result.
which is the desired result. □
3 Main results
In this section we present the pointwise convergence rate and the uniform convergence rate of to its extreme value distribution under different normalizing constants. The first result is the pointwise convergence of extremes under the normalizing constants given by (1.2).
for large n, where , are defined in (1.2).
Recently Liu and Fu  proved the result, we present an improved proof for the pointwise convergence rate in Section 4.
The following is the uniform convergence rate of extremes under the appropriate normalizing constants and given by (1.3) and (1.4), which shows that the optimal convergence rate is proportional to .
where and are defined by (1.3) and (1.4), respectively.
where . By similar arguments, we can obtain (4.2).
The proof is complete. □
Together with (4.12), we establish (4.7).
Hence (4.8) is proved.
which is (4.9).
This is (4.10). The proof is complete. □
The research was supported by the National Natural Science Foundation of China (11171275) and the Fundamental Research Funds for the Central Universities (XDJK2012C045).
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