Warren has hit on the most logical culprite - a straight reactor core type HID Ballast.
This is very commonly used for low power H.P.S. lamps, driven from a 120 VAC source. Since the lamp is 55 VAC, the Ballast / Lamp combination works well at 120 VAC without the need of an Autotransformer... Until...
The straight Reactor Ballast will respond Big-Time to fluctuations at 5% and above. This works out to drops / rises of Voltage in the range of 6 VAC on the 120 VAC system becoming a major influence to the light output or operation. Voltage sags in the area of 15% to 20% will cause a H.P.S. lamp to drop out very easilly.
The CWA [Constant Wattage Autotransformer] can withstand Voltage sags of 30-40% before the lamp's plasma dies off [lamp goes out].
The CWA is somewhat linear to fluctuations - 10% [+/-] change in line voltage usualy results in a 5% change in lamp intensity.
CW [Isolated Transformer] Ballasts have the best lamp regulation.
As far as the Capacitor goes, it's only used with the straight Reactor Ballast to bring the Power Factor up from 50% to near 90%. It performs no other function than this.
The same goes for High Reactance [HX] type Autotransformer Ballasts.
When a Capacitor is used in the Secondary of a CWA / Reg-lag Ballast, it improves Power Factor and is also used to "Tweek" the lamp's performance.
From the sounds of things, I would say the Voltage / Current at the HPS lamp comes very close to zero for maybe 1/8 cycle - or drops below 40 VAC for upto 4 cycles when the lamp extinguishes. On Fluorescent lamps, the transient drop is noticed as a brief "Pulse" or "Flash", because they recover rapidly as compared to HID lamps.
Incandescent lamps will do the "Dim then Surge, then recover" deal. They are noticable during the event from the transient drop in light output, which is followed by a rapid overshoot in lamp intensity. This makes the event seem much different than it really is.
HIDs cannot lose their plasma, then instantly re-establish it, so when the plasma becomes too thin, the lamp goes out.
The starting arc cannot be maintained across the arc tube until the tube's temperature is low enough to drop the pressure inside the arc tube.
When the pressure inside the tube drops low enough to allow enough current to be pushed by the available level of Voltage [to jump between the Electrodes] - this will create the initial starting arc and sustain it long enough to produce a nice fat plasma!
Pardon the strange descriptions above