| The lamp discharge present a negative dynamic resistance - as increased current yield to lowered voltage (in 1..100ms timescale range - when the ion density is capable to change, but notthe temperature.). This creates, with conjunction with a capacitor, relaxing oscillator - the capacitor charges, till the arc build, this yielad to increased current, this to lowering the arc voltage, what causes even more current, yielding to sudden discharge of the capa by huge current spike. When the voltage falls, the arc extinguishes, so all current from the ballast is then charging back the capacitor and the cycle repeats.
When the capacitor is too small, it discharges sooner, then the ionisation density changes, it's stable. When too high value is connected, the recgarge time is so long, the ionisation and (and for even higher the electrode temperature as well) falls too much, so the arc is not able to restrike anymore.
As these oscillation cause different power dissipation distribution among lamp parts (electrodes, arc radiation, arc heat) then the steady mode, temperatures in the lamp change. But these temperature changes might lead to change in arc behavior, co condition for those oscillation disappear, yielding original dissipation spread, so temperatures "return" to their original values, where the oscillation starts again. So the steady state and oscillation state cycle, so if one state is brighter then the other, it might demonstrate as flashing lamp.
But current surges when hose oscillation happen, are very aggressive to lamp electrodes and cause their accelerated wear, so lamp blackening and/or premature emitter failure.
That's why only limited capacitance is allowed with mercury lamps and none (long cable, as distributed LC, does not yield such current spikes) with other HID types. And generally not using any such capacitor is recommended as a default approach.
And why cheap fluorescent lightsticks (and/or nightlights) with only a capacitor as ballast impedance cause so short lamp life.
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