Since his first wife vanished more than three decades ago, Robert A. Durst, the eccentric and estranged son of one of New York’s most prominent real estate dynasties, has lived under the suspicious gaze of law enforcement officials in three states.

They have followed his path from New York City to Los Angeles, where one of his closest friends was found dead in her home in 2000. They have tracked him to Galveston, Tex., where he fled after investigators reopened the case of his wife’s disappearance, and where he posed as a mute woman and shot and dismembered a neighbor in 2001.

[Read about what happens next in the Robert Durst case.]

Mr. Durst was acquitted in the Texas killing, and was never arrested in the disappearance of his wife or the death of his friend. But on Saturday, he found himself in custody once again, arrested on a charge of murder as he walked into a New Orleans hotel he had checked into under a false name.

On Sunday night, in the final moments of the final episode of a six-part HBO documentary about him, “The Jinx: The Life and Deaths of Robert Durst,” Mr. Durst seemed to veer toward a confession that could lift the shroud of mystery that surrounds the deaths of three people over the course of three decades.