"He would walk early in the morning – around 6:30 a.m. or 7," says retired city-worker Tony Rivera, 52, who walks his dog in the park each day at that time, "because, he said, he always had trouble sleeping. That's why he'd come out so early in the morning."

Once in a while, Ledger, 28, would watch the early-morning chess players in action – sometimes even joining them. He often talked about his daughter, says Rivera, who struck up a chat with the actor during one of his sleepless strolls. "He said it was a great thing, being a dad."

But Ledger's friendly banter with park locals was always punctuated by one recurring complaint: "He'd say, 'I'm very tired,' " Rivera recalls, "and he looked it."

Ledger's reported hard-partying drinking at all hours of the night may have been another way for him to try and self-medicate himself to sleep. "He partied a lot," one friend of Ledger's told PEOPLE. "He didn't really stop partying. At nights when I was going home, he was just starting. He was part of the New York scene." Another club regular recalls that Ledger would arrive at Manhattan lounge, The Beatrice Inn, at 3:30 a.m., "and he never left until sunrise."