Nevermind, I thought you meant G(t) was price but I guess you mean it is a predictor variable. Maybe regress ATR on realized G's?
If G is truly gaussian and ATR is dependent on G, then the best estimate for the next value of ATR is the mean of ATR, just as the best guess for the next value of G is m. With random variables you have to go with the means. But in real life ATR is not a random walk and I doubt if there is a gaussian variable upon which it is dependent.
another perspective is that atr oscillates around a base due to mean reversion. if atr is > or < base then the prediction should be skewed to the oposite. the simple formula is atr/close resulting in an indicator which somewhat behaves as an oscillator. The idea here is the dominant market pattern is expansion/contraction/expansion/contration etc. Gotta get back to making pizza and delivering in 30...