Nops...........the situation is as follows:-
1) The CFLs should remain in OFF condition as the respective switches of them are in OFF state.
OK, but in response to a previous question you told me that the relays controlling the CFLs are turning on and off.
So are they? Or are they always off, and the CFL turns on anyway?
2) The only switch that is in ON state is that of fan.
I assume you mean relay, not switch? And therefore you confirm that the relay contacts for the CFLs remain open.
3) To control the fan speed , we have connected "step type fan regulator which is having 5 steps) in series to the fan supply.
I'm not familiar with "step type fan regulators". Is this a separate controller you switch to the correct speed?
4) Now to vary the speed , we vary the fan regulator steps, as we do so , the CFLs also turns ON-OFF.
So as you manually do stuff with your "step type fan regulator", nothing at all happens in the circuits you've shown me (i.e. no change in relay states, or anything else) but the lamps go on and off?
5) So my confusion is that why CFLs turns ON-OFF as I try to vary fan speed, Why they don't remain in permanent OFF state.
Well, as I see it, we still don't have enough circuit diagram to show how the CFLs and the fan are connected, so I will simply assume that the CFLs are connected via your relay contacts in parallel with the fan and its controller.
Do the lamps emit a brief weak flash, or do they fully turn on. If the latter, for how long?
It's not unusual to see a CFL bulb emit a brief weak flash as you connect or disconnect it from an otherwise dead circuit. I presume this is due to capacitive effects. It is possible that switching the fan causes sufficient inductive effects to tickle the bulbs into emitting a weak flash.
If you tell me that the lamps come on full brightness (or significantly bright) and that the relay contacts are open, then I will be somewhat amazed. It could be that you have a live/neutral fault and that you are seeing some sort of capacitive coupling through to earth, but... well, let's hear what you say.