It is actually less likely that the children of the top 1% will have a better standard of living than their parents, than that the children of the middle or lower percentages will. The parents in the top 1% have a standard of living it would be hard for their children to top.
Disclaimer: Being a child of a parent in the top 1% certainly makes it more likely that one will also be in the top 1%. But even in good times, such a child falling out of the top 2 or 3% will experience a fall in living standards relative to their parents.
The question wasn't whether their kids would be in the top 1%, it was whether or not they'd have a better standard of living.
To the extent that having a good life comes from one's personal income, if the purchasing power of the dollar were increasing then they could easily buy more even if they were only in, say, the top 10%.
But most of what comprises standard of living isn't even determined by one's personal income, but rather by the resources, laws, customs, etc. of the community at large. And as it stands the wealth and health of society as a whole is dropping so fast that an individual can pretty much never keep up, no matter how much money they make or where they choose to live.
Disclaimer: Being a child of a parent in the top 1% certainly makes it more likely that one will also be in the top 1%. But even in good times, such a child falling out of the top 2 or 3% will experience a fall in living standards relative to their parents.