The conditions necessary for sustained oscillation can be summarized as follows:
1. There should be a positive feedback.
2. The loop gain of the circuit must be equal to or greater than 1 i.e. Aβ<=1.
3. The phase shift around the circuit must be zero or 360 degrees.
The non-sinusoidal oscillation at steady state can happen at 2 states:
1. β is not exactly 1.
2. Due to saturation.
And α decimal signal or noise voltage can provide a measurable output voltage and the circuit can act as oscillator even without an input signal.