论文信息 - The Growth of Random Walks and Levy Processes

The Growth of Random Walks and Levy Processes

Let {Xi} be a sequence of independent, identically distributed non-degenerate random variables taking values in Rd and Sn = Iq=I Xi, Mn = MaXi-isn I Si. Define for x > 0, G(x) = P XI 1> x}, K(x) = x-2E(X, 12 XI 1 XI c x)), M(x) = I E(X, 1 { I XII c x}) 1, and h(x) = G(x) + K(x) + M(x). Then if fi = sup {a lim sup xah(x) = 01, i = sup {a lim inf xah(x) = 01, it is proved that n-'/aMn -s 0 for a (, while the lim inf is 0 and the lim sup is 00 for fi < a < S. Some alternative characterizations of the indices A, 8 are obtained as well as the analogous results for Levy processes.

William E. Pruitt | W. Pruitt

[1] W. Pruitt. General One-Sided Laws of the Iterated Logarithm , 1981 .

[2] M. Klass. Precision Bounds for the Relative Error in the Approximation of $E|S_n|$ and Extensions , 1980 .

[3] M. J. Klass,et al. Toward a universal law of the iterated logarithm , 1976 .

[4] W. Pruitt. The Hausdorff Dimension of the Range of a Process with Stationary Independent Increments , 1969 .

[5] Bert Fristedt,et al. Sample Functions of Stochastic Processes with Stationary, Independent Increments. , 1972 .

[6] N. Jain,et al. Maxima of partial sums of independent random variables , 1973 .

[7] P. Hartman,et al. On the Law of the Iterated Logarithm , 1941 .

[8] C. Esseen. On the concentration function of a sum of independent random variables , 1968 .

[9] N. Jain,et al. The Other Law of the Iterated Logarithm , 1975 .

[10] William Feller,et al. A Limit Theorem for Random Variables with Infinite Moments , 1946 .