Centered Occupation Variables

• Centered occupation variables are defined as occupation times with the mean subtracted and often normalized to analyze stochastic fluctuations precisely.
• They enable rigorous study of limit theorems and asymptotic behaviors, revealing both Gaussian and non-Gaussian scaling limits in various processes.
• Applications include renewal theory, Lévy processes, and branching systems, aiding in parameter estimation and simulation of occupation dynamics.

A centered occupation variable is a stochastic process or random variable derived by subtracting the expected value (mean) from an occupation time variable, typically followed by a normalization. The occupation time of a set $A$ up to time $t$ for a Markov process or a particle system is the total measure (usually Lebesgue) of times the process spends in $A$. Centering allows rigorous paper of fluctuations, limit theorems, and the probabilistic structure of occupation times, especially in regimes where scaling limits produce nontrivial Gaussian or non-Gaussian processes. This concept is fundamental in renewal theory, Lévy process analysis, branching particle systems, and statistical parameter estimation.

1. Definition of Centered Occupation Variables

Occupation time for a set %%%%3%%%% up to time $t$ is defined as

$\alpha(t) = \int_0^t 1_{X(s) \in A} \, ds,$

where $X(s)$ is the underlying process. The centered occupation variable is given by subtracting the mean:

$$\tilde\alpha(t) = \alpha(t) - \mathbb{E}[\alpha(t)].$$

In scaling limits or fluctuation studies, normalization by an appropriate factor $H_T$ is typical:

$$X_T(A;t) = H_T^{-1} \left( \int_0^{Tt} Z(s,A) \, ds - \mathbb{E} \int_0^{Tt} Z(s,A) \, ds \right),$$

where $Z(s,A)$ is the number of particles or state indicator in a system at time $s$ and $H_T$ is a scaling chosen relative to the variance growth (López-Mimbela et al., 2023, Starreveld et al., 2016).

2. Asymptotics and Fluctuations

For alternating renewal processes where sojourn times in $A$ and its complement $B$ are i.i.d. pairs $(D,U)$, the expected occupation time grows linearly:

$$\mathbb{E}[\alpha(t)] = \frac{\alpha}{\alpha+\beta} t + O(1),$$

with $\alpha = \mathbb{E}[D], \beta = \mathbb{E}[U]$.

The variance grows linearly as well:

$\operatorname{Var}[\alpha(t)] = v t + o(t),$

where

$$v = \frac{\beta^2 \sigma^2_\alpha + \alpha^2 \sigma^2_\beta - 2\alpha\beta c}{(\alpha + \beta)^3},$$

with $\sigma^2_\alpha = \operatorname{Var} D$, $\sigma^2_\beta = \operatorname{Var} U$, $c = \operatorname{Cov}(D,U)$ (Starreveld et al., 2016).

A central limit theorem holds:

$$\frac{\alpha(t) - m t}{\sqrt{v t}} \Rightarrow N(0,1) \quad \text{as} \quad t \to \infty,$$

where $m = \alpha/(\alpha + \beta)$. This establishes that the rescaled, centered occupation variable converges in distribution to a Gaussian random variable, encapsulating the standard fluctuation regime (Starreveld et al., 2016).

In critical branching systems with migration, the occupation variable (centered and rescaled) exhibits convergence to non-Markovian Gaussian processes—sub-fractional Brownian motion and its weighted variants, depending on the lifetime and migration parameters—revealing rich fluctuation structures beyond standard CLTs (López-Mimbela et al., 2023).

3. Analytical Characterizations and Transforms

Laplace–Stieltjes transforms of centered occupation variables provide integral representations and facilitate the analysis of processes up to random times (e.g., exponentially distributed epochs):

$$\int_0^\infty e^{-q t} \mathbb{E}[e^{-\theta\alpha(t)}]\, dt = \frac{1}{1 - L_{1,2}(q+\theta, q)} \left[ \frac{1 - L_1(q+\theta)}{q+\theta} + \frac{L_1(q+\theta) - L_{1,2}(q+\theta, q)}{q} \right],$$

with $$L_1(\theta) = \mathbb{E}[e^{-\theta D}], \quad L_{1,2}(\theta_1, \theta_2) = \mathbb{E}[e^{-\theta_1 D - \theta_2 U}]$$ (Starreveld et al., 2016).

For spectrally positive Lévy processes reflected at the infimum, joint Laplace transforms involve scale functions $W^{(q)}, Z^{(q)}$ and the right-inverse of the Laplace exponent $\psi$ (Starreveld et al., 2016).

Large-deviation asymptotics for the occupation time are also obtainable using cumulant generating functions

$$\Lambda(\theta) = \lim_{t \to \infty} \frac{1}{t} \log \mathbb{E}[e^{\theta \alpha(t)}],$$

and the rate function $$\Lambda^*(q) = \sup_\theta [\theta q - \Lambda(\theta)]$$, so that for $q > m$,

$$\lim_{t\to\infty} \frac{1}{t} \log \mathbb{P}(\alpha(t)/t \geq q) = -\Lambda^*(q).$$

The root $d(\theta)\in(0,1)$ satisfying $\mathbb{E}[ e^{ \theta(1-d) D - \theta d U } ] = 1$ characterizes the transform (Starreveld et al., 2016).

4. Centered Occupation Variables in Branching Systems

For age-dependent critical branching particle systems with symmetric $\alpha$-stable migration, fluctuation limits depend on the regime of the lifetime distribution:

• Finite-mean lifetimes ($\mathbb{E}S < \infty$, $\alpha < d < 2\alpha$): The normalized occupation time converges to sub-fractional Brownian motion (sub-fBm) with parameter $h = 3 - d/\alpha$, and explicit covariance

$$\operatorname{Cov}(\zeta^h(s), \zeta^h(t)) = s^h + t^h - \frac{1}{2}\left[(s+t)^h + |s-t|^h\right].$$

• Pareto-type lifetimes ($0<\gamma<1$, $\alpha\gamma < d < \alpha(1+\gamma)$): The covariance kernel is

$$Q(s, t) = \frac{1}{1-b} \int_0^{s\wedge t} r^{\gamma-1} \left[ (s-r)^b + (t-r)^b - (s+t-2r)^b \right] dr,$$

with $b=2-d/\alpha$, characterizing a weighted sub-fractional Brownian motion (López-Mimbela et al., 2023).

These centered fluctuation limits, always zero-mean Gaussian, display path continuity, self-similarity, and, for some parameters, long-range dependence and non-Markovianity, as detailed by explicit covariance and scaling relations.

5. Centered Occupation Variables in Continuous Processes

For the centered Ornstein–Uhlenbeck process, the occupation time below a fixed threshold $a$ up to time $T$ is

$$\tau_a(T) = \int_0^T \mathbf{1}_{\{X_t \le a\}}\, dt,$$

which, centered and possibly normalized, captures fluctuations around the mean time spent below $a$. Explicitly, for $X_t$ solving $dX_t = -\lambda X_t dt + \sigma dW_t$,

$$\mathbb{E}[\tau_a(T)] = \int_0^T \Phi\left(\frac{a}{\sqrt{k(t)}}\right)\,dt,$$

with $$k(t) = \frac{\sigma^2}{2\lambda}(1 - e^{-2\lambda t})$$, and $\Phi(\cdot)$ the standard normal CDF (Bock et al., 2011).

Least squares approaches for parameter estimation use the discrepancy between measured occupation times and theoretical expectations to infer drift and volatility coefficients, leveraging the link between mean occupation times and process parameters.

6. Applications and Computational Aspects

Centered occupation variables underpin limit theorems in the paper of random walks, renewal and Lévy processes, and particle systems. In reliability theory, the occupation time in an "up" state offers availability analyses; in finance and physics, they measure time intervals for threshold crossings or regime persistence (Starreveld et al., 2016, Bock et al., 2011).

Simulations combine pathwise Monte-Carlo with analytic error function integrals for validation and parameter estimation, requiring careful numerical handling of singularities and variance control through high sample sizes (e.g., $N\gtrsim10^4$) (Bock et al., 2011).

Weighted sub-fractional Brownian motion, emerging as a fluctuation limit in systems with heavy-tailed lifespans, provides a framework for studying processes exhibiting both self-similarity and non-Markovian memory effects with explicit covariance characterization (López-Mimbela et al., 2023).

7. Properties and Structural Features

Centered occupation variables and their limit processes possess:

• Self-similarity: Scaling properties determined by the exponent in the normalization; for weighted sub-fBm, ${\zeta(c t)} \equiv_d c^{(a+b+1)/2} \zeta(t)$.
• Path regularity: Hölder-continuity exponents are regime-dependent.
• Long-range dependence: For certain parameter ranges, covariance decay is slow, implying persistent correlation.
• Non-Markovianity: Centered fluctuation limits like (weighted) sub-fractional Brownian motions fail the covariance factorization property required for Markov processes (López-Mimbela et al., 2023).

These properties are essential in determining the analytical tractability, simulation strategies, and limiting behavior of models used in stochastic physics, mathematical biology, and applied probability.

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References:

• "Occupation times of alternating renewal processes with Lévy applications" (Starreveld et al., 2016)
• "Parameter Estimation from Occupation Times" (Bock et al., 2011)
• "Occupation time fluctuations of an age-dependent critical binary branching particle system" (López-Mimbela et al., 2023)