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Determining computational complexity of stochastic processes

I have some points $(x_i,y_i)$ generated by a program. These values are not exact, but are random fluctuations around an average value (though quite close, I think). I want to know whether $y(x)=O(e^x)$ or $y(x)=O( poly \,\, x)$. Does anyone know how, and to what degree of certainty, this distinction is possible?

Note: This question is an alternative formulation of my question Determining computational complexity of stochastic processes.

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