Erdoğan, M. Burak2016-01-082016-01-081995http://hdl.handle.net/11693/17671Ankara : Department of Mathematics and The Institute of Engineering and Science of Bilkent University, 1995.Thesis (Master's) -- Bilkent University, 1995.Includes bibliographical references leaves 44-45We prove that the function 1 , a 6 (0 ,2 ), ^ e R, 1 + is a characteristic function of a probability distribution if and only if ( a , 0 e P D = {{a,e) : a € (0,2), \d\ < m in (f^ , x - ^ ) (mod 27t)}. This distribution is absolutely continuous, its density is denoted by p^(x). For 0 = 0 (mod 2tt), it is symmetric and was introduced by Linnik (1953). Under another restrictions on 0 it was introduced by Laha (1960), Pillai (1990), Pakes (1992). In the work, it is proved that p^{±x) is completely monotonic on (0, oo) and is unimodal on R for any (a,0) € PD. Monotonicity properties of p^(x) with respect to 9 are studied. Expansions of p^(x) both into asymptotic series as X —»· ±oo and into conditionally convergent series in terms of log |x|, \x\^ (^ = 0 ,1 ,2 ,...) are obtained. The last series are absolutely convergent for almost all but not for all values of (a, 0) € PD. The corresponding subsets of P D are described in terms of Liouville numbers.i, 45 leavesEnglishinfo:eu-repo/semantics/openAccessCauchy type integralCharacteristic functionCompletely monotonicityLiouville numbersPlemelj-Sokhotskii formulaUnimodalityQA273.6 .E73 1995Distribution (Probability theory).Functions, characteristic.Thesis