---
title: Generalized Trigonometric Densities
url: https://www.emergentmind.com/topics/generalized-trigonometric-densities-gtds
type: topic
---

# Generalized Trigonometric Densities

Generalized Trigonometric Densities (GTDs) constitute a three-parameter family of univariate probability densities constructed using Drábek–Manásevich generalized trigonometric functions. GTDs subsume classical laws such as the Gaussian, stretched-$q$-Gaussian, hyperbolic secant, logistic, and raised-cosine densities as special cases. This family admits closed-form normalization, systematically interpolates between compact support, exponential, and power-law tails, and provides extremal solutions to entropy–information trade-offs under generalized moment constraints. GTDs provide a rigorous framework for characterizing heavy-tailed phenomena and optimizing generalized information-theoretic quantities [2410.24141].

## 1. Formal Definition and Analytical Structure

Let $p>0$ and set $p^* = \tfrac{p}{p-1}$. For real parameters $\beta, \lambda$, impose the conditions
\[
1+\beta-\lambda \neq 0, \quad \text{and} \quad \mathrm{sign}(1-\lambda+\beta) = \mathrm{sign}\left(\frac{1-\lambda}{p}+\beta\right).
\]
Define the "stretch" parameter
\[
\kappa_{p,\beta,\lambda} = \left| \frac{\lambda-1}{1+\beta-\lambda} \right|^{1/p^*}, \quad v = \frac{\lambda-1}{\lambda-\beta}, \quad w = p^*.
\]
The GTD is then given by
\[
c_{p,\beta,\lambda}(y) = a_{p,\beta,\lambda} \times
\begin{cases}
\bigl[\cos_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda > 1), \\[2ex]
\bigl[\cosh_{v,w}(\kappa_{p,\beta,\lambda} y)\bigr]^{1/(\lambda-\beta)}, & (\lambda < 1),
\end{cases}
\]
where $\cos_{v,w}$ and $\cosh_{v,w}$ are generalized cosine and hyperbolic-cosine functions. The normalization constant is
\[
a_{p,\beta,\lambda} = \frac{p^* \kappa_{p,\beta,\lambda}}{2}\, B\left( \frac{1}{p^*}, \frac{\beta\,\mathrm{sign}(1-\lambda+\beta)}{|1-\lambda|} + \frac{1}{p} 1_{\{1-\lambda>0\}} \right).
\]
The domain of definition $X_{p,\beta,\lambda}$ guarantees normalization $\int_{X_{p,\beta,\lambda}} c_{p,\beta,\lambda}(y)\,dy = 1$ using the generalized Pythagorean identity (for trigonometric or hyperbolic case as appropriate) [2410.24141].

## 2. Relation to Classical and Generalized Laws

The GTD construction encompasses a broad collection of standard and generalized probability densities:

- **Stretched $q$-Gaussian:** For $\beta = \lambda$, the density reduces to the generalized stretched Gaussian:
  \[
  c_{p,\lambda,\lambda}(y) = a_{p,\lambda}\, \exp_{2-\lambda}(-|y|^{p^*}),
  \]
  where $\exp_q$ denotes the $q$-exponential, widely used in nonextensive statistical mechanics.

- **Logistic law:** For $(p,\beta,\lambda) = (2,1,0)$, one obtains, up to normalization, $\mathrm{sech}^2(y)$.

- **Hyperbolic secant law:** $(p,\beta,\lambda) = (2,0,-1)$ produces $c(y) \propto \mathrm{sech}(y)$.

- **Raised-cosine law:** $(p,\beta,\lambda) = (2,3/2,2)$ yields $(\cos y)^2$ on $[-\pi/2, \pi/2]$.

This unifying structure allows analysis and interpolation between compactly supported kernels, sub-Gaussian, and heavy-tailed regimes, all within the same parameterization [2410.24141].

## 3. Generalized Entropic and Information-Theoretic Properties

The GTDs are extremal solutions for a variety of information-theoretic quantities:

- **Rényi entropy:** For order $\alpha>0$, $\alpha \ne 1$,
  \[
  H_\alpha[f] = \frac{1}{1-\alpha} \log \int f(x)^\alpha\,dx.
  \]

- **Generalized Fisher information:** Following Lutwak and Bercher, define
  \[
  \phi_{p,\lambda}[f] = \left( \int |f^{\lambda-2}(x) f'(x)|^p f(x)\,dx \right)^{1/(p\lambda)}.
  \]

A fundamental Stam-type inequality holds: for $p \ge 1$, $\beta > 0$, $\lambda > 1-\beta p^*$,
\[
\phi_{p,\beta}[f]\;\exp(H_\lambda[f]) \ge \phi_{p,\beta}[c_{p,\beta,\lambda}]\; \exp(H_\lambda[c_{p,\beta,\lambda}]).
\]
Fixing $\phi_{p,\beta}[f]$ shows that the GTDs $c_{p,\beta,\lambda}$ uniquely minimize $H_\lambda$, achieving minimal Rényi entropy at fixed generalized Fisher information [2410.24141].

## 4. Deformed Cumulants and Generalized Moments

GTDs provide a natural operator for the construction of generalized moments based on deformed cumulative distributions:

- **Deformed CDF:** For order $\gamma$,
  \[
  F_\gamma(x) = \int_{-\infty}^x f(t)^\gamma\,dt.
  \]

- **Generalized moment of order $r$ (cumulative form):**
  \[
  \mu_{r,\gamma}[f] = \int_{-\infty}^{+\infty} |x|^r\,dF_\gamma(x) = \int_{-\infty}^{\infty} |x|^r f(x)^\gamma\,dx.
  \]
  This can also be written as
  \[
  \mu_{r,\gamma}[f] = \int_{-\infty}^{\infty} \left|\int_0^x f(t)^{1-\gamma}\,dt \right|^r f(x)\,dx,
  \]
  with the deformation parameter $\gamma$ tuning the weight placed on the distribution's tails. These moments reduce to those of a deformed/regularized distribution and are key to the characterization of heavy-tailed behavior [2410.24141].

## 5. Heavy-tail Interpolation and Critical Finiteness Thresholds

The GTD family achieves a continuous interpolation between compact support, exponential decay, and power-law tails:

- For $f(x) \sim |x|^{-\eta}$ as $|x| \rightarrow \infty$, all $\mu_{r,\gamma}[f] < \infty$ for every $r > 0$ if and only if
  \[
  \gamma < \gamma_c = \frac{1}{\eta^*}, \qquad \eta^* = \frac{\eta}{\eta-1}.
  \]
  This threshold marks the point at which the deformation parameter $\gamma$ sufficiently suppresses the tails to ensure finiteness of moments. Thus, $\gamma_c$ can be interpreted as the phase transition for moment finiteness in the presence of heavy tails. For $\gamma < \gamma_c$, the density is compactly supported; at $\gamma = \gamma_c$, moments become critical with exponential tails; for $\gamma > \gamma_c$, moments diverge and the law exhibits power-law tails [2410.24141].

## 6. Fundamental Properties and Optimization Principles

The structure of GTDs endows the associated generalized moments with several key properties:

- **Monotonicity in $r$:** For fixed $\gamma$, the function $r \mapsto \sigma_{r,\gamma} := \mu_{r,\gamma}^{1/r}$ is nondecreasing (by Hölder's inequality).

- **Scaling behavior:** For scaling $x \mapsto \kappa x$, the moment rescales as
  \[
  \mu_{r,\gamma}[f_\kappa]^{1/r} = \kappa^{-1} \mu_{r,\gamma}[f]^{1/r}
  \]
  where $f_\kappa(x) = \kappa f(\kappa x)$.

- **Extremality:** Maximizing $H_\lambda[f]$ under constraints $\mu_{r,\gamma}[f] = M$ and $\int f = 1$ leads, via the Lagrange and Euler–Lagrange formalism, to a solution of the GTD type:
  \[
  f(x)^{\lambda-1} \propto \alpha \left|\int_0^x f(t)^{1-\gamma}dt\right|^r + \beta.
  \]
  The matching minimization of generalized Fisher information achieves a duality between entropy and information for this family.

These properties ensure that GTDs act as attractors or optimal distributions under a wide class of entropic and moment constraints [2410.24141].

## 7. Summary Table: GTDs and Special Cases

| Family                | GTD Parameters                        | Tail Type     |
|-----------------------|---------------------------------------|---------------|
| Stretched $q$-Gaussian| $\beta = \lambda$                     | Exponential/Power|
| Logistic              | $(2,1,0)$                             | Exponential   |
| Hyperbolic Secant     | $(2,0,-1)$                            | Exponential   |
| Raised Cosine         | $(2,3/2,2)$                           | Compact       |

GTDs unify these and interpolated behaviors in a single closed-form parameterization [2410.24141].

---

Generalized Trigonometric Densities provide a unified analytic and information-theoretic framework for continuous probability densities, covering a broad range of decay types, optimizing entropy-information inequalities, and supporting rigorous characterization of heavy-tailed, sub-Gaussian, and compactly supported behavior [2410.24141].

Source: https://www.emergentmind.com/topics/generalized-trigonometric-densities-gtds