By T. A. Azlarov, Ya. M. Husainov (auth.), Gisiro Maruyama, Jurii V. Prokhorov (eds.)
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Additional resources for Proceedings of the Third Japan — USSR Symposium on Probability Theory
Of R ~ ( P-a)= PI l r ( ~ ) 4cdt). ,, ) with ~(U)--~ u . Hence where ~ is a ~- measure on ~ , and (8) implies that of 5g The last formula establishes a one-to-one correspondence betwee~ all the homogeneous sures ~ ~-functionals of (~(Y),p) and all the on E which charge no sets inaccessible this assertion, we need only shew that, if which are inaccessible to ~ ~ oC~ . To prove charges no sets , then the measure has the same property relative to to ~ ~-mea- ~ defined by (9) . It can be checked with the help of the results of Section 5 9 Now we consider an arbitrary process (X~,~CI),~) , satisfying the conditions of Theorem 2.
Almost all sample functions of the stationary Gaussian random process ~p, belong to the space p~oo , if and only if and hence if K for all large Will a ~ . 367-378). not, sums The last theorem reminds these results. Its proof will be omitted and published elsewhere. We show now how to derive Theorem 5 from the last result and results of Section I. ~ > S } ~ ' - ~ ( 2"" ' , where as above Hence ~2 and the inclusion ~ H ~ H; follows from Theorem 6. To prove necessity, note that and -- cE( I (l~ct)l-E -~ Q -~I(Ecl~/t)i- g l~Lct)l)2//;dr = ((i-9) R~co)) .
Nected with the ~arkov process be the semigroup of ~ . )(Ws } >. Hence ~ CdX~)2~C2ds. sfxWs~l=ds. becomes a one-dimensional Brownian motion, and the nonramdom moment 5 =~ becomes a Markov time bounded by C # . I Indeed, Of c o u r s e , Corollary t h e same i n e q u a l i t y 2. }. 27 where M is the median. This inequality will be used in Section 3. Note also that I M-rnl'- C. A question of interest is to investigate how essential is the Gaussian property of the measure ~ for Theorem I. We give a gene- ralization of this theorem for a non-Gaussian case.