On a class of unimodal distributions

Laha, R. G.

doi:10.1090/s0002-9939-1961-0121826-3

Cited by 26 publications

(22 citation statements)

References 1 publication

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“…These densities can be viewed as generalizations of symmetric Linnik's densities. For // = 1, \0\ = m in(^,7r -^) , these densities had appeared in the papers by R. G. Laha [10] and R. N. Pillai [11]. Therefore, the problem of study of analytic and asymptotic properties of the densities with characteristic function (1.2) seems to be of interest.…”

Section: Ostrovskii and A Haj'favi [8] Carried Out A Detailed Investmentioning

confidence: 99%

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

Erdoğan

1999

The Journal of Fourier Analysis and Applications

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Section: Ostrovskii and A Haj'favi [8] Carried Out A Detailed Investmentioning

confidence: 99%

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

Erdoğan

1999

The Journal of Fourier Analysis and Applications

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“…Every semi-stable (or stable) distribution belongs to the domain of partial attraction (or the domain of attraction) of itself. It is known 245 that a distribution G belongs to the domain of partial attraction of the semi-stable distribution F corresponding to (2) if and only if for some [C,,}, C~>0, and for an increasing sequence I(n) of positive integers, ( 4 ) lim…”

Section: ( 3 ) G*~(b~x + A)mentioning

confidence: 99%

On the domain of partial attraction of semi-stable distributions

Shimizu¹

1970

Ann Inst Stat Math

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A characteristic function 9(t) (or the corresponding distribution F) is said to be semi-stable, if for some constants T>I and c>1, 9(t)=9~(c-lt),for all * holds. Every semi-stable characteristic function is infinitely divisible. A complex valued function 9(t) defined on the real line is a non-normal semi-stable characteristic function which satisfies (1) converges (or contains a subsequence which converges) weakly to F, where asterisk denotes the convolution operator. Every semi-stable (or stable) distribution belongs to the domain of partial attraction (or the domain of attraction) of itself. It is known 245

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“…The properties of the Linnik distributions were studied in many papers. We should mention [6,25,26,31] and other papers, see the survey in [19].…”

Section: The Linnik Distributionsmentioning

confidence: 99%

On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems

2020

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We present new mixture representations for the generalized Linnik distribution in terms of normal, Laplace, exponential and stable laws and establish the relationship between the mixing distributions in these representations. Based on these representations, we prove some limit theorems for a wide class of rather simple statistics constructed from samples with random sized including, e. g., random sums of independent random variables with finite variances and maximum random sums, in which the generalized Linnik distribution plays the role of the limit law. Thus we demonstrate that the scheme of geometric (or, in general, negative binomial) summation is far not the only asymptotic setting (even for sums of independent random variables) in which the generalized Linnik law appears as the limit distribution.

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On a class of unimodal distributions

Cited by 26 publications

References 1 publication

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

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

On the domain of partial attraction of semi-stable distributions

On Mixture Representations for the Generalized Linnik Distribution and Their Applications in Limit Theorems

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