IEFERENCES [11 13. V. GNEI)ENKO and A. N. KOLMOGOROV, Limit distributions [or sums o[ independent random variables, GITTL, Moscow-Leningrad, 1949. (English translation by K. L. Chung, Addison-Wesley, 1954.) [2] E. K. LEBEDINTSEVA, On limit distributions ]or normalized sums o[ independent random variables, Dokl. Akad. Nauk USSI, No. 1, 1955. (In Russian.) ON A HYPOTHESIS PROPOSED BY B. V. GNEDENKO V. M. ZOLOTAREV (MOSCOW) and V. S. KOROLYUK (KIEV)Several years ago Academician t3. V. Gnedenko proposed the following:Let n (l/Bn)(1+ +n)--'/In be a sequence of normed sums of independent stochastic quantities having a nondegenerate limit distribution G(x) for appropriately selected constants A and B n. If among the distributions of stochastic quantities e there are only s different ones, then the limit distribution G(x) is a composition of not more than stable laws.In the paper the hypothesis proposed by 13. V. Gnedenko is proved for s 2 and an example is presented showing that the theorem by E. Lebedintseva [21 does not prove this hypothesis in its entirety.In Ell and [21, formulas are given for the mean number of crossings of some level by a stationary Gaussian process, and in [31 a formula is given for the mean number of zeros, but without rigorous proof. In [4] a rigorous proof is given for a formula for the mean number of zeros in the rather special case of the Gaussian process N (t) E aj(Xj cos 2jt+Yi sin 2t), =1 where the X and Y are independent Gaussian variables with mean 0 and dispersion 1.We shall start by proving a general assertion which does not require the assumption that he process is Gaussian.Theorem 1. I! the one-dimensional density o/the process (t) is bounded and the derivative ' (t) is continuous, with probability 1, then the number o/crossings o/the level u on the segment [a, bl by the process (t) is/inite, with probability 1, and moreover, the probability that (t) becomes tangent to the level u is zero.

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