There is a good correlation between the wealth of the parent and the wealth of the children in many countries, including the US.

In economics this is called "social mobility" and it is measured to analyze the success of economic and social policies, whereby, generally, it is considered that having a high social mobility index is a desirable thing in a country. That's because a country with high social mobility is a country where the American dream is happening more often...

Social mobility works both ways: in a country with high social mobility, it is more likely that a child will be much poorer, or much richer than the parents.

There is a paper by the OECD analyzing by how much different factors (family wealth, education, etc.) affect wealth of children across different countries: http://www.oecd.org/centrodemexico/medios/44582910.pdf