In this note we prove a generalisation of a conditioned functional central limit theorem of Bolthausen (cf. ). This generalisation explains the nature of the discontinuity between such conditioned limit theorems for random walks (or in queueing for waiting times) with or without drift.
"Conditioned Limit Theorems and Heavy Traffic." Ann. Probab. 12 (2) 631 - 634, May, 1984. https://doi.org/10.1214/aop/1176993310