Linnik distribution¶
A symmetric heavy-tailed probability distribution whose characteristic function has Linnik form.
Core Idea¶
The Linnik distribution is a symmetric geometric-stable probability distribution, commonly centered at zero, with characteristic function \(\varphi(t)=1/[1+(\lambda|t|)^\alpha]\) for scale \(\lambda>0\) and stability index \(0<\alpha\le2\) under a standard parameterization. It is “geometric stable” because an appropriately normalized sum of an independent geometrically distributed number of identically distributed variables can retain or converge to the same family. This parallels ordinary stable laws, where stability is defined for a fixed number of summands, but changes the compounding count from fixed to geometric. The index controls tail weight.
Scope of Application¶
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Geometric random sums. A geometrically distributed number of contributions motivates the distribution's aggregation structure.
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Heavy-tailed modeling. Symmetric data with an appropriate tail index can be compared with Linnik rather than assumed Gaussian.
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Characteristic-function inference. Transform-based fitting and diagnostics are natural where the density lacks a convenient closed form.
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Mixture representations. Alternative stochastic constructions support simulation and theoretical calculation.
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Numerical density and distribution evaluation. Inversion methods require parameterization, truncation, and accuracy checks.
Clarity¶
Linnik distribution names a specific symmetric geometric-stable family, commonly identified by characteristic function \([1+(\lambda|t|)^\alpha]^{-1}\) under a stated parameterization. It is not the ordinary stable law with the same index, and scale conventions must be declared before comparing parameters. The term makes the random geometric number of summands central to the stability property.
Manages Complexity¶
The Linnik distribution compresses a heavy-tailed symmetric law into a scale parameter, stability index, and a simple rational characteristic function. The analyst reads tail weight and moment existence chiefly from the index, recognizes the Laplace boundary at the appropriate endpoint, and uses geometric-stability structure to handle random sums. Parameterization conventions form a necessary branch for scale comparison.
Abstract Reasoning¶
Identification move. From a fitted or limiting characteristic function of the standard rational form, infer a symmetric Linnik law only after fixing scale and index conventions. Tail move. From the stability index, infer relative tail heaviness and which ordinary moments fail to exist. Compounding move. Use geometric random-sum structure to derive closure or convergence results distinct from fixed-sum stable laws. Boundary move. Similar heavy tails or symmetry do not identify the family; verify the transform or an equivalent representation. Special-case move.
Knowledge Transfer¶
Within the home domain. The Linnik distribution transfers across probability theory, heavy-tailed modeling, geometric-stable limits, and stochastic processes wherever its characteristic function and parameterization define the law. Tail behavior, symmetry, scale, stability relations, and mixture representations retain formal meanings. Beyond the home domain (C — probability model). It applies literally to any data-generating context for which the distribution is a defensible model; subject matter can change without analogy.
Relationships to Other Abstractions¶
Current abstraction Linnik distribution Domain-specific
Parents (1) — more general patterns this builds on
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Linnik distribution is a kind of Probability Distribution Domain-specific
Linnik distribution is a domain-specific kind of Probability Distribution: A symmetric heavy-tailed probability distribution whose characteristic function has Linnik form.
Hierarchy paths (5) — routes to 3 parentless roots
- Linnik distribution → Probability Distribution → Random Variable → Function (Mapping)
- Linnik distribution → Probability Distribution → Probability → Measure → Set and Membership
- Linnik distribution → Probability Distribution → Probability → Measure → Aggregation → Micro Macro Linkage
- Linnik distribution → Probability Distribution → Random Variable → Probability → Measure → Set and Membership
- Linnik distribution → Probability Distribution → Random Variable → Probability → Measure → Aggregation → Micro Macro Linkage
Neighborhood in Abstraction Space¶
Linnik distribution sits in a sparse region of the domain-specific corpus (69th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Probability Transforms & Tail Behavior (7 abstractions)
Nearest neighbors
- Yule–Simon Distribution — 0.85
- Bernstein inequalities (probability theory) — 0.85
- Cramér's Theorem (Large Deviations) — 0.85
- Kolmogorov's Three-Series Theorem — 0.84
- Monotone Likelihood Ratio Property — 0.83
Computed from structural-signature embeddings · 2026-10-08