By Buldygin V. S.
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Extra resources for Skorokhod's works related to investigations of probability distributions in infinitely dimensional spaces
Example text
79) which implies that the exponential c1 multiplying it is regularly varying. 11. 79). The second, linearly independent solution y2 is then obtained from y2(x) = yi(x) f exp(— f g(s)ds)dt. 79) and the Representation theorem, this leads to y2(x) = x f where L3(t) = Lr2(t) with some E(s) —+ 0, as s —+ oo. The above integral diverges due to Proposition 4 (ii), and since < 1. By linear equations 43 0, one obtains applying Proposition 1, and observing Proposition 7 with That L2 is normalized and the necessity part of the proof for Y2 follows Y2.
If cbt is slowly varying, then for all there holds positive decreasing solutions y(x) of equation e f A remark analogous to Remark 1 ,3 + o(1))ds). holds also here. Proof. 5: =— + But the integrand is a regularly varying function of index —1. 2 (with replaced by —or) decreasing solutions are of the form y(s) = > 0, and due to the assumption of the theorem, ç1(s) is slowly varying and tends to zero and one has in addition f(s) = ç1(s)s2 + cs2. 10) (with replaced by —cr), so that c. By Proposition 7, the function (sa(s))F is regularly varying of index —1.
By this we mean as usual: to find an ultimately positive and continuous function 'p such that y(x)/'p(x) —+ 1, as x —+ oo. 1), the properties of regularly varying function is the only one essential tool. In principle the same procedure applies for both slowly and regularly varying solutions. However, for the latter case the asymptotic representation becomes too involved. This is why the results for these are presented in less general form then for the former case. The results of this Chapter are based on [29J, [42J and [19J.
Skorokhod's works related to investigations of probability distributions in infinitely dimensional spaces by Buldygin V. S.
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