In normal linear regression analysis, many model selection rules proposed from various viewpoints are available. For the information criteria AIC, FPE, $C_p$, PSS and BIC, the asymptotic distribution of the selected model and the asymptotic quadratic risk based on each criterion are explicitly obtained.
"Asymptotic Properties of Criteria for Selection of Variables in Multiple Regression." Ann. Statist. 12 (2) 758 - 765, June, 1984. https://doi.org/10.1214/aos/1176346522