Gerunds can take either the genitive (his-less common/more formal) or the accusative (him-more common/less formal) as subject or object. When the possessive alternative is used, it usually focuses attention on the action described in the ‘-ing’ clause.
What do you think about him or his buying a new car? - ¿Qué te parece que compre/haya comprado un coche nuevo?
I wasn't happy about them or their asking him to leave - No me pareció bien que le pidieran/hubieran pedido que se fuera
I don't like their acting - No me gusta su actuación
I don't like them acting - No me gusta que actúen
"His conviction was based on his having had sexual relations with a 14-year-old girl." Washington Post-Oct 31, 2016
"Seagal's close involvement with Russia led to his being dropped from the line-up of an Estonian music festival in July 2014" Telegraph.co.uk Nov 3 , 2016
"Yet his trying to assist in the identification of a mass murderer is perfectly understandable." Herald Scotland-Nov 1, 2016
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