My family moved into our home 10 years ago. The people we bought the house from only had a dog when we bought the place and yet I found a litter box in the closet upstairs and some cat food. We had 4 cats so I cleaned out the closet good in case there were fleas.
Soon after we moved in, I would be in the closet hanging up clothes, and hear one of our cats run up the stairs. Or so I thought. I'd look out of the closet but nothing would be there. This happened a lot and still happens.
Then one night I was lying in our bed trying to sleep, when I felt one of the cats jump up on the bed. I had my back to the door so I didn't see which one it was. It walked behind my back and lay down. I looked around to see which of our cats it was but there was nothing there.
We adopted another cat that was a stray and I brought him into our room until the other cats could get use to him. He was fine on our bed, but refused to get on the floor. He is still very cautious coming into our room. He'll look all around the floor before he dares to come in. He creeps slowly through the bedroom door, jumps up on the chair by the door, then on to our bed to get in the room.
I'm not afraid of our ghost cat. I now welcome it when I feel it jump up on the bed. I just say, "Hello, ghost kitty. Are you going to sleep with me?" It walks up beside me and lies down, and I'll fall asleep.