Asymptotic equivalence between the default Bayes factors and the ordinary Bayes factors with intrinsic priors

In Bayesian model selection or testing problems, default priors are typically improper; that is, the resulting Bayes factor is not well defined. To circumvent this problem, two methodologies, namely, intrinsic and fractional Bayes factors are proposed and developed. Further, these two Bayes factors are asymptotically equivalent to the ordinary Bayes factors computed with proper priors called intrinsic priors. However, it seems that there are some necessary conditions to satisfy asymptotic equivalence. Such conditions are derived and justified in this article and illustrative examples are provided. Simulations are performed to demonstrate the results.