Assuming that you could produce the financial records to prove the exact arrearage amount (and hopefully your mother kept a detailed journal every week, or month with exact amounts and dates) and assuming there is a court order for support, your mother's estate has a right to collect any debt owed to it, including child support.
If she left a will, the executor named in that will, after being sworn in by the court, could attempt, likely by lawsuit, to collect.
If there was no will, the administrator of her estate could pursue the amounts. You might seek to become one once you turned 18.
Before doing anything, show a lawyer what records you have, any will, and other pertinent information to determine if this is worth spending legal fees on. If your mother couldn't collect, it may be harder to collect now. Let a lawyer assess the chances and go from there.
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I agree with Mr. Ashman's response. You'll want to take whatever details and documentation you have to consult with an attorney. However, without knowing the facts of your mother's estate, distribution of the estate is a big question. Did your mother have a will, will and a trust, are you a beneficiary under those documents or were you disinherited in those documents, if they exist? There are a lot of details that you need to review with local counsel for guidance.
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