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Who can claim a deceased person's property?
Their heirs. Either through their will (whether probated or not -- if it exists and still can be probated it needs to be) or through intestate succession. The Affidavit of Heirship will suffice if they had no will. Keep in mind that Texas law isn't as straightforward on succession as you would think. If there is a surviving spouse and kids, there is separate and community property. If there is an ex- and step kids, well, I don't even want to think about that.
Funny story: in law school, during our estates and trusts class, an Aggie PhD math guy (another student) and I gave a math refresher lecture to those who needed help with the math involved with succession exercises. Up until that point, I was embarrassed at my less than stellar performance in Math 151 at A&M (at the time, Engineering Calc I; don't know if it still is). After that, I felt like a math genius, comparatively speaking.