Central limit theorem on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">CAT</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> spaces with contracting isometrie

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> be a group acting on a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">CAT</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> space with contracting isometries. We study the random walk generated by an admissible measure on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>G</mml:mi> </mml:math> . We prove that if the action is non-elementary and under optimal moment assumptions on the measure, the random walk satisfies a central limit theorem. The general approach is inspired from the cocycle argument of Y. Benoist and J-F. Quint, and our strategy relies on the use of hyperbolic models introduced by H. Petyt, A. Zalloum and D. Spriano, which are analogues of the contact graph for the class of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi mathvariant="normal">CAT</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> spaces. As a side result, we prove that the probability that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> th-step the random walk acts as a contracting isometry goes to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>1</mml:mn> </mml:math> as <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> goes to infinity.