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The mortgage-application process can be daunting for even the most creditworthy borrower. But what about foreign applicants, who may not have a documented credit history or permanent residence in the U.S.? What hurdles do noncitizens face when applying for a jumbo mortgage?

Foreign investment in U.S. residential real estate skyrocketed to a new high between April 2016 and March 2017, according to the National Association of Realtors, a real-estate trade group. During that time, foreign buyers purchased $153 billion of residential…