If X is an asymptotic normal variate then we can find the asymptotic distribution of h(X) by using delta method if h(X) \in R. But if h(X) is not Real valued function (e.g., h(X) could be a positive function), what is the asymptotic distribution of h(X)?
The function h(.) does not have to be real-valued. It only requires h'(.) to exist and its value to be non-zero. Either use delta method with complex derivative or use transformation method for probability densities (assuming h(.) is invertible).
Asymptotic in what sense? I presume you mean that the Central Limit Theorem is applicable to this distribution. If not I am not sure what you are asking or saying. Also when you say "not real" my classical mathematical interpretation of what you are mean is that x has an imaginary number component "i"; where i^2=-1. If h has an imaginary component it can not be observed in the real world. I think you should more clearly define what you mean by "real". I think you might mean h(x) is a "positive-definite" function. In this case, you are talking about a left-truncated distribution function. If you are trying to generate an empirical smoothed & fitted PDF for the observed data, both normality & truncation are irrelevant - just pump the data into a standard procedure. On the other hand if it is mu & sigma you are after, you need to either assume truncation does not significantly alter mu & sigma or you might try a log transformation & look to see if the transformed distribution is "sort-of" normal like, then you can compute mu & sigma & assume it a log-normal distribution.
Dear Karim
See
https://faculty.washington.edu/ezivot/econ583/asymptoticsprimer.pdf
https://arxiv.org/pdf/1207.5899.pdf
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