An analytical representation of a random process with independent increments in some space (random walks) is considered. Using L.P.Kadanoff terms about the unit increment as one hop, the law of walked particles distribution in the space is derived from the general representation of stochastic elementary hops (hop’s probability distribution law).  There is a natural way to obtain an ordinary diffusion (known S.Chandrasekhar’s result) for limited hops and hops which have the distribution with all moments. There appear also the P.Levy random walks (or Levy flights) for hops distribution laws without the second moment. At last, there are truncated Levy random walks while the hops distribution has moments larger than the second one. The exact asymptotical form of truncated Levy walks distribution is derived.

            Obtained results compared with known empirical data for return fluctuations of international assets and stock indexes (inverse cubic law [1]) as well as for Russian stocks and RTS index (the same inverse cubic cumulative distribution for returns). The independent increments for Russian assets return are verified by autocorrelations: very short (7 minutes) for RTS index, and practically zero for stocks. The appearance of transition to Gauss fluctuations for one month RTS return is probably detected.

            Thus, all investigated assets demonstrate the general law: the unit hop cumulative distribution is inverse cubic. There is a good agreement of the elaborated theory with empirical data.

[1] P.Gopikrishnan, M.Meyer, L.A.N.Amaral, and H.E.Stanley. Eur. Phys. J. B. 3, 139 (1998); P.Gopikrishnan, V. Plerou, Y. Liu, L.A.N. Amaral, X. Gabaix, H.E. Stanley. Physica A. 287, 362 (2000).

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