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The validity of our Monte Carlo simulation procedure (the time sequentialprocedure) had been verified by the corresponding analytical procedure which isindependent of our method methodologically. Also, the results obtained by ourprocedure are compared with those obtained by the different Monte Carlosimulation procedure (the v_{cut} procedure) which has been exclusivelyutilized by different authors and the agreement between ours and theirs arefound to be good. By utilizing our Monte Carlo procedure, the validity of whichis guaranteed in two different procedures, we investigate not only thefluctuation of high energy muons themselves but also the fluctuation of thevarious quantities related to the energy losses by the muons, which aredifficult to obtain by the v_{cut} procedure. Namely, we obtain the fluctuationon energy losses of muons in the present paper(Part1). In a subsequent paper(Part2), based on the idea developed in Part1, we will examine the correlationbetween the primary energies of muons and their energy deposits and thecorrelation between the energy deposits of the muons concerned and theirCherenkov light yields in the KM3 detector in detail.

A detailed study on the range fluctuations of 10^{11}eV to 10^{18}eV  muons in water and the fluctuations of the Cherenkov light yields due to the  accompanied cascade showers initiated by muons. Part1: High Energy Muons