## Analytic and asymptotic properties of non-symmetric Linnik's probability densities

##### Author

Erdoğan, M Burak

##### Advisor

Ostrovskii, Lossif V.

##### Date

1995##### Publisher

Bilkent University

##### Language

English

##### Type

Thesis##### Item Usage Stats

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Show full item record##### Abstract

We 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.

##### Keywords

Cauchy type integralCharacteristic function

Completely monotonicity

Liouville numbers

Plemelj-Sokhotskii formula

Unimodality

QA273.6 .E73 1995

Distribution (Probability theory).

Functions, characteristic.