Digital is easiest on paper. Uncorrelated random values give you white noise up to the Nyquist frequency = 1/(2*sampling interval). The probability distribution of the random values does not matter, only that they're uncorrelated/independent.
Yeah, but you'd need a LFSR (or similar method) running at 10GHz+*, to achieve the stated bandwidth. (On the upside, an LVDS or ECL output would be quite simple to couple into a 50 ohm transmission line, and deliver a modest number of dBm's.)
*As I recall, it's not flat up near the sample rate, due to time correlation (it's only random every t = n*Ts, not T ~= 2*pi/BW). The result is a sinc(f) weighting, which flattens out at lower frequencies (say < Fs/5 or 10) to a quite reasonable white noise signal.
A smaller bandwidth is probably a smart idea, anyway. A quite reasonable bit of noise can be made easily (say, 100s of MHz), which could then be modulated up to higher frequencies using a diode balanced mixer and a variable frequency generator (as local oscillator). This, hopefully, will give a flat band with a small peak in the middle (due to LO bleed-through).
Any amplification following the mixer will be rather expensive, because 2GHz (with a lower limit of ~DC??) isn't the cheapest thing. Not with calibrated gain anyway. (MMICs are cheap, sure, but they aren't very flat either.)
Tim