I just re-curated this profile and was going to add a citation for her birth date and place, but I encountered something unusual.
There's no California birth index showing in either Ancestry or MyHeritage for a Kimberly Kardashian. I also went through the Los Angeles County birth index for 21 October 1980 with no luck -- 462 babies born that day, none with that name or anything similar. I also checked surrounding years and still came up empty.
It looks like all the reputable encyclopedias give birth date as 21 October 1980 and her birthplace either as Los Angeles or Beverly Hills. Her daughter North's birth certificate also gives that date and a birthplace of California.
So I guess we're left to assume that she somehow escaped indexing or has had her birth index legally shielded (which would be extraordinarily rare, and it's hard to think of why that would happen in this case).
Either way, figured I'd make a note in case others run into this same issue. I've set the birthplace as the more generic Los Angeles County since I can't prove Beverly Hills, which is what the profile previously said.
(FWIW, at least two of her siblings are indeed indexed.)
