Before revision:  V-transforms describe the relationship between quantiles of the return distribution and quantiles of the distribution of a predictable volatility proxy variable constructed as a function of the return. V-transforms can be represented as copulas and permit the formulation and estimation of models that combine arbitrary marginal distributions with linear or non-linear time series models for the dynamics of the volatility proxy.
After revision:  V-transforms describe the relationship between quantiles of the return distribution and quantiles of the distribution of a predictable volatility proxy variable . They can be represented as copulas and permit the formulation and estimation of models that combine arbitrary marginal distributions with linear or non-linear time series models for the dynamics of the volatility proxy.