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Independent And Stationary Sequences Of Random Variables - Chapter 20

Chapter 20 SOME UNSOLVED PROBLEMS In this chapter we list various unsolved problems and possible lines of further research, classified according to the chapters from which they arise. They come from various sources, to which it would be difficult to give exact credit . | Chapter 20 SOME UNSOLVED PROBLEMS In this chapter we list various unsolved problems and possible lines of further research classified according to the chapters from which they arise. They come from various sources to which it would be difficult to give exact credit. Chapters 3-5 1 The problem of extending the results of 3.4 to the case of convergence to a stable law of exponent a 2 see 191 21 . Thus we would wish to find necessary conditions and sufficient conditions not too far apart for the distribution function F x of the normalised sum of independent identically distributed random variables in the domain of attraction of the stable law Ga x to satisfy F x -Ga x 0 n 7 0. The method of 3.4 does not work since Theorem 1.6.1 is not applicable. 2 What is the multi-dimensional analogue of Theorem 3.4.1 3 In the notation of 3.5 write Cn sup sup F x - Z x . F x P3 Theorem 3.5.2 says that C 10 i 3 6 27r i as n- oo It would of course be interesting to know Cn explicitly but failing that to find estimates of C perhaps the second term of an asymptotic expansion. 4 Along the same lines as the last problem set C x lim sup sup F x i x . n-oo F P3 It has been conjectured by Kolmogorov 82 that for symmetric distributions F Chap. 20 SOME UNSOLVED PROBLEMS 391 C x 2n -ie -x2. 20.1.1 This has been proved under restrictive conditions by Linnik 98 . 5 Find an analogue of Theorems 4.5.1 and 4.5.3 for the case of convergence to a stable law with exponent a 2. It seems possible that this problem is simpler than 1 . 6 How is Prohorov s theorem 4.4.1 changed if convergence in Lt oo oo is replaced by convergence in Lp oo oo l p oo 7 Again in the spirit of 1 extend the results of 5.3 to the case of convergence to stable laws. Chapters 6-14 1 In all the theorems of these chapters the zones considered were of width A n p n or A n p n where p n is a function increasing arbitrarily slowly to infinity. Can the function p n be replaced by a constant say 1 Many results in this direction have been