Aram does not actually tell us anywhere in this story what his earliest memories were. The story itself represents a memory from "the good old days" when he was a nine year old child, and still felt the world to be a place full of mystery and imagination and life to be a "mysterious dream." He remembers the day when his cousin rapped on his window and revealed that he had picked up a white horse from somewhere. The two boys then keep this horse until the owner of the horse comes looking for it and identifies the horse as his. Although he does not accuse the boys of having stolen the horse, he makes clear to them that he knows the horse is his. This causes the boys to be either guilt-stricken or fearful of being revealed, and they return the horse to its owner under cover of darkness.
The family themselves have earlier memories than this—the boys' uncle Khosrove tells stories about "the old country" in the parlor of the house. But Aram does not tell us anything about his life before the white horse arrived, except that his family was poor and that the whole tribe, in fact, was "poverty stricken."