Multivariate $\alpha$-normal distributions

The Weibull distribution can be obtained using a power transformation from the standard exponential distribution. In this article, we will consider a symmetrized power transformation of a random variable with the standard normal distribution. We will call its distribution the $\alpha$-{\it normal (Gaussian) distribution}. We examine properties of this distribution in detail. We calculate moments and consider the moment problem of $\alpha$-normal distribution. We derive the formula of its differential entropy and (exponential) Orlicz norm. % of $\alpha$-normal random variables. Moreover, we define the joint distribution function of the multivariate $\alpha$-normal distribution as a meta-Gaussian distribution with $\alpha$-normal marginals. We consider also the limiting distribution as $\alpha$ tends to infinity.