Gerunds are verbal forms (-ing) acting as nouns, naming actions, implying actors. The actors are in many cases not stated. But when they are, they demand a possessive pronoun (my, his), not an objective personal pronoun (me, him).

Consider this sentence from a leader in this week's Economist on an Iranian student protester who escaped to America:

The regime did not want to be blamed for him dying behind bars, he says, so he was allowed out for treatment.

The gerund dying demands the possessive his in the phrase: for his dying behind bars.

The preposition for seduced the writer into using him rather than his.

The gerund dying demands the possessive his in the phrase for his dying behind bars because the object of the preposition for is dying. In the Economist example, him is the object of the preposition for; dying behind bars becomes a participial phrase modifying him. Imagine the (illogical) sentence without the nonessential participial phrase:

The regime did not want to be blamed for him, so he was allowed out for treatment.

I used to see the error only occasionally. Now I see it almost always.

A good test is to plug in first-person pronouns, both objective (me) and possessive (my), and to see how the gerund phrase works:

The regime did not want to be blamed for me dying behind bars...

The regime did not want to be blamed for my dying behind bars...

The first sentence is unidiomatic, the second grammatical.

In some sentences the possessive is superfluous because the gerund's actor is obvious, as in a sentence I heard last night on the News Hour:

By him showing up, he signaled to the attendees at the NAACP convention that he would listen to their concerns and consider their advice.

By showing up, he signaled to the attendees at the NAACP convention that he would listen to their concerns and consider their advice.