By Yoshikazu Baba (auth.), G. Maruyama, Yu. V. Prokhorov (eds.)

ISBN-10: 3540063587

ISBN-13: 9783540063582

ISBN-10: 3540469567

ISBN-13: 9783540469568

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Lo~ve: Probability Theory, Van Nostrand, 1960. Steklov ~athematical Institute of the Academy of Sciences of the USSR Mosoow ON THE G E N E R A T I O N OF M A R K O V BY S Y M M E T R I C Masatoshi I. PROCESSES FORMS Fukushima Introduction The a d v a n c e of the t h e o r y of M a r k o v processes during 15 years was p r i n c i p a l l y led by the n o t i o n property. Hunt b u i l t up a p r o b a b i l i s t i c Among theory based process lished others, on a c e r t a i n (cf. M. K. of t r a n s f o r m a t i o n s strong M a r k o v p r o c e s s of the strong M a r k o v strong M a r k o v p r o c e s s Blumenthal named Getoor potential now c a l l e d [2]) process a Hunt and D y n k i n of a s l i g h t l y a standard the past more (cf.

18 (1970), 73-84. Moran: On asymptotically optimal tests of composite hypotheses, Biometrika, 57, 1 (1970), 47-55. Nagaev: Some limit theorems for large deviations, Teor. , lO, 2 (1965), 231-254. Neyman: Optimal asymptotic tests of composite statistical hypotheses, Probability and statistics, Uppsala, Almquist and 45 Wiksells, (The Harald Cram@r Volume), 1959, 213-234. Yurinsky: Bounds for characteristic functions of certain degenerate multidimensional distributions, Teoro Verojatnost. , 17, 1 (1972), 99-110.

32) If ~(~1 i I ('°J)(X~ p(~i d ~ is vector-valued then ~E~ e means that this condition is fulfilled for each component. In particular, if ~&~i , fE ~i , or #% ~, then, respe- ctively, ~ (o,f~ moreover, ~ p / ~ ~ p(x, %,a) -=(~##+ ~; )p where ~# -- E#e , then 35 We shall not give sufficient conditions for the possibility of differentiations under the integral sign. Rather these and other relations themselves obtained by the differentiation under the integral sign will be used as conditions of theorems.

### Proceedings of the Second Japan-USSR Symposium on Probability Theory by Yoshikazu Baba (auth.), G. Maruyama, Yu. V. Prokhorov (eds.)

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