Yes, people seem to forget that all past immigrant and refugee groups came over with nothing, lived in extremely poor areas for a generation or two, before making it out of poverty. Immigrants don't come over with already established wealth, and the vast majority do come here to make a better life for themselves. Wealth doesn't come to someone in a year or two either, and in most cases, the older parent immigrants never do make it out of poverty, and it's the children who begin to see an increase in wealth.
The Italians, Jews, Chinese, all pulled themselves out of poverty. Hell, the very first european immigrants to these shores were in relative terms were extremely poor, and it took several generations before even their descendants made a name for themselves.