You are
"fairly sure" but
very wrong. Let us clarify (all of the following is about
French PhD programmes).
1. The criteria for selecting a student for a PhD are publicly established by each doctoral school. They do not require any justification of income/wealth.
2. PhD students are not required to sign any contract or to be paid any salary.
2.1. Some of them have a "doctoral contract" (a new thing created in 2009 specially for PhD students), some are public servants, some have signed regular employment contracts of three years specifically for the PhD. However, a large part do not get any benefit from the doctoral school: they are either unemployed, or have a job unrelated to their PhD.
2.2. The director of the doctoral school is supposed to check that all conditions, including financial ones, are met so the student can work well.
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Addendum:
- At the beginning of the 2000s, 30% of students dropping out of their PhD said it was "for financial reasons" [1].
- PhD students are almost always enrolled in a university, or in an institution related to the doctoral school. This is about 500€/year if you are lucky enough to be French, but can go up to 7000€/year if you are from outside the EU.
- Some institutions, for instance some Écoles des Mines or the École Polytechnique, still pay PhD students an "informal salary" without any employment contract [2]. This is in a grey zone, legally speaking.
* * *
[1] Philippe Moguérou, Jake Murdoch, Jean-Jacques Paul. "Les déterminants de l’abandon en thèse", 10es Journées d'études Céreq, 2003-05-23. http://web.archive.org/web/20070928014312/http://www.cereq.f...
[2] Anne Jouan. "Des doctorants étrangers travaillent encore au noir", Le Figaro, 2008-10-20. http://www.lefigaro.fr/formation/2008/10/20/01015-20081020AR...