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Lipari and Stanev developed a method for range fluctuation of high energymuons, stressing the importance of accounting for the fluctuations of theenergy loss in radiative processes in 1991 and, now, their method has becomethe basement for the energy determination of high energy muons through themeasurement of the Cherenkov light yields due to those muons in KM3 physics.Once, Takahashi et al. developed a method for the investigation on the depthintensity relation of high energy muons in which all the stochastic processesconcerned are taken into account exactly (1983). Now, we make the method byTakahashi et al. revival for the same purpose of the application to theanalysis of future KM3 physics. In the present paper, our concern is restrictedto the introduction to the fundamental of our method and some subsequentresults thereby in which the real simulated behaviors of high energy muons from$10^{12}$eV to $10^{18}$eV, the survival probabilities of high energy and so onare included. The discussion around the practical application of our method tothe KM3 physics is entrusted in the subsequent papers. As far as the survivalprobability of high energy muons is concerned, our method gives nearly the sameresults to Lipari and Stanev's in some regions and gives the deviated resultsfrom theirs in another ones. Thus, we examine the application limit of theirmethod and clarify the reason why, comparing with our method. The most distinctdifference in the both methods may become apparent in the treatment on theCherenkov light yields spectrum by which one may estimate the energies of themuons concerned. We will mention to them in subsequent papers.

A fundamental theory based on the Monte Carlo \textit{Time Sequential Procedure} for the range fluctuations of high energy muons