Bochner'sRepresentationTheorem

Bochner's representation theorem is a fundamental result in the theory of characteristic functions, which are used to study random variables and stochastic processes. An L-stable process is a particular type of stochastic process that has a specific form of the characteristic function, exhibiting stable distributions. These processes are often used to model heavy-tailed phenomena in various fields like finance, physics, and engineering.

Bochner's representation theorem states that for any continuous positive definite function, there exists a unique Borel measure such that the function can be represented as the Fourier transform of that measure. In the context of an L-stable process, the characteristic function is given by:

$$\Phi(t) = \mathbb{E}[\exp(i \cdot t \cdot X)] = \exp(-c |t|^{\alpha} \cdot (1 + i \cdot \beta \cdot \text{sign}(t) \cdot \tan(\frac{\pi \alpha}{2})))$$

Here, the parameters are as follows:

• $t$ is a scalar argument representing time.
• $X$ is the random variable associated with the L-stable process.
• $\alpha$ is the stability parameter ($0 < \alpha \leq 2$), which characterizes the tail behavior of the distribution. When $\alpha = 2$, the process reduces to a Gaussian process.
• $\beta$ is the skewness parameter ($-1 \leq \beta \leq 1$), which measures the asymmetry of the distribution.
• $c$ is a scale parameter ($c > 0$), which influences the overall scale of the distribution.
• $\mathbb{E}$ denotes the expectation operator, and $\exp(x)$ is the exponential function $e^x$.
• $i$ is the imaginary unit ($\sqrt{-1}$), and $\text{sign}(t)$ is the sign function.

The Bochner's representation of the L-stable process is essentially a way of expressing the process's characteristic function in terms of its stable distribution parameters. This representation is crucial in understanding the behavior of L-stable processes, their statistical properties, and their application in various domains.