I live in a small town with very few homeless people, I've only seen 2. But one really stands out.
He has a bit of a story to him. He was a manager of a factory for almost 25 years, but one day he was hit in the head by a machine, and had permanent damage done to his mind.
So he was fired, but that's not what he's known for, after the accident he started cheating on his wife, I suppose the whole "I'm working late" doesn't work when you're unemployed. So anyways they got a divorce, and she gained custody of the house or something, or maybe he just left, I'm not totally sure. But after he left he started to wander the town, he never begs for money, he just... Roams. You can find him hanging outside of a McDonalds talking to himself, sometimes he gets food, but he doesn't eat it, he saves it for later, I've never seen him actually eat in the places he gets food.
He's known as Slippery Bob because he cheated on his wife after his on the job accident, and then his life was pretty much ruined...
But he always talks to himself, and stares at my mom's ass so I'd say he's kinda like the description you gave.