However there has been an obvious shift in the last 15 years, to the extent that the old 3 generation rule has been changed. This has been primarily due to certain members of the nouveau rich accumulating enough wealth to more than match (and in many cases exceed) many of the Old Money crowd. Thus currently Oprah would gain entry into the Old Money crew (which was always a 100% WASP group), and other members of the new rich (for example Hedge fund manager Steve Cohen who have personal yearly incomes between 600-1 billion dollars) can gain entry anytime they please.
Thus I guess traditionally it was always about inherited wealth that had to have a lineage line of at least 3 generations affixed to it, and currently it has shunted to basically include the traditional Old money group PLUS any nouveau riche that has sufficient funds.
Actually there was a time when the traditional definition meant a connection to a royal bloodline of some sort, so it seems the definitions do change.
It's more complicated than that -- there are intersections between old wasp money and old jewish money (yes, there is such a thing) in that world. For instance, a Guggenheim would have no problems getting into one of those apartments. There is also old Italian money, etc, However, there are places where even old money has trouble cracking the elite, such as Fisher Island, etc.
A good deal depends on how the "new money" is earned. Respectable professions like banking, medicine, or law will turn the family in a shorter time than entertainment (vide your Oprah example). Hoteliers and restaurateurs are doubtful. My grandmother would still be calling the Hiltons "nouveaux riches" (although only under her breath to close friends).
Her family was very poor when she was growing up, living in a shotgun house on a side street in Augusta, GA . . . but she was descended from the brother of a Signer and one of the first families of Charleston SC as well. No amount of poverty will erase the cachet of being from an old Charleston family (BTW the family fortune was lost when her grandmother's father and mother died in a yellow fever epidemic, and the estate was consumed caring for the minor children). She married well and started that side of the family back up to prosperity. Oddly enough, she was college educated and a bacteriologist at the Medical College of Georgia in Augusta before she married.
This is why I love rappers.
They are all capitalists (mostly Christian), and they can buy and sell anybody.
They are creating a whole generation of capitalists and conservatives, and the Left doesn't even know it yet.