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The validity of our Monte Carlo simulation procedure ({\it the integralmethod}) had been verified by the corresponding analytical procedure of whichis quite independent of our method methodologically. Also, the results obtainedby our procedure are compared with those obtained by the different Monte Carlosimulation procedure ({\it the differential method}) which have beenexclusively utilized by the different authors and the agreement between themare found to be well. By utilizing our Monte Carlo procedures, the validity ofwhich is guaranteed in two different procedures, we investigate not only thefluctuation of high energy muons themselves but also fluctuation of the variousquantities related to the energy losses by the muons, which are difficult toobtain by {\it the differential method}. Namely, we obtain fluctuation onenergy losses of the muons, fluctuation on Cherenkov lights due to theaccompanied cascade showers initiated by the muon and the correlations betweenthem . Finally, we obtain the transition curves for Cherenkov lights in KM3detector, taking into account of all possible fluctuations in the stochasticprocesses and point out the difficulty of the reliable estimation of the energyof the muons which are resultants of muon neutrino events in the KM3 detectors.

A Detailed Study on the Range Fluctuations of $10^{11}$eV to $10^{18}$eV  Muons in Water and the Fluctuations of the Cherenkov Lights due to the  Accompanied Cascade Showers initiated by Muons