Asymptotic Distributions of Record Values under Exponential Normalization

In this paper, we study the limit distribution of the record values under nonlinear normalization of the form T-n(x) =exp{u(n)(vertical bar log vertical bar x vertical bar vertical bar)(vn) sign(log vertical bar x vertical bar) }sign(x), which is called exponential norming (e - norming). The corresponding limit laws of the upper extremes are called e-max stable laws (denoted by U(.)). In this paper, we show that the limit distributions of the record values under exponential norming are of the form N(- log( - log U(x))), where N ( .) is the standard normal distribution. Moreover, we study the domains of attraction for these types of limit laws. Finally, some illustrative examples are given.