Truncated Variation Functional in fBm

• Truncated Variation Functional quantifies fBm's sample path regularity by summing increments above a given threshold, generalizing classical total variation.
• It exhibits sharp exponential integrability and distinct phase transitions at H=1/2, with different concentration behaviors for sub- and super-diffusive regimes.
• Advanced techniques like chaining and metric entropy link truncated variation to 1/H-variation and local times, providing insights into fBm's intricate path structure.

The truncated variation functional offers a critical tool for quantifying the regularity of sample paths of fractional Brownian motion (fBm) by controlling the magnitude and frequency of path oscillations at a fixed truncation scale. For fBm—a family of centered Gaussian processes parameterized by the Hurst index $H\in(0,1)$—the truncated variation functional admits a precise probabilistic and analytic description, including sharp concentration, moment, and asymptotic results. These properties exhibit a phase transition at $H=1/2$ and connect to broader stochastic path properties such as $1/H$-variation and local times.

1. Formal Definition and Properties

Let $B^H=(B_t^H)_{t\ge0}$ be an fBm of Hurst index $H\in(0,1)$. The truncated variation on $[0,T]$ at truncation level $\delta>0$ is defined as

$$TV^\delta(B^H)_{0}^{T} := \sup_{\Pi:0=t_0<\cdots<t_n=T} \sum_{i=1}^n \left(|B_{t_i}^H - B_{t_{i-1}}^H| - \delta \right)_+,$$

where $(x)_+ = \max\{x,0\}$ and the supremum is over all partitions $\Pi$ of $[0,T]$.

This functional generalizes total variation, as it measures the sum of those increments of $B^H$ in the partition whose magnitude exceeds the threshold $\delta$. Unlike total variation, $TV^\delta(B^H)$ remains finite for typical sample paths of fBm for all $H$.

For any real-valued function $f:[0,T]\to\mathbb{R}$, the truncated variation also admits the characterization

$$TV^c(f;[0,T]) = \inf\left\{ TV(g;[0,T]) : \|f-g\|_\infty \leq \frac{c}{2} \right\}$$

with the infimum attained, where $TV(g;[0,T])$ denotes the total variation of $g$ on $[0,T]$ and $\|\cdot\|_\infty$ the uniform norm. This definition is equivalent to the partition supremum for $fBm$ and, in particular, for any continuous $f$.

One-sided truncated variations are given by

$$UTV^\delta(B^H)_0^T := \sup_{\Pi} \sum_{i=1}^n \left(B^H_{t_i} - B^H_{t_{i-1}} - \delta\right)_+,$$

$DTV^\delta(B^H)_0^T := UTV^\delta(-B^H)_0^T,$

with $$TV^\delta(B^H)_0^T = UTV^\delta(B^H)_0^T + DTV^\delta(B^H)_0^T$$.

2. Exponential Integrability and Moment Generating Functions

The law of $TV^\delta(B^H)_{0}^{T}$ possesses distinctly different tail behaviors depending on the value of $H$.

For $fBm$ with $H\in(0,1)$, there exist constants $A_H,B_H,C_H<\infty$ such that for all $u\geq0$ and all $\delta,T>0$,

$$\Pr\Bigl(TV^\delta(B^H)_{0}^{T} \geq \delta^{\frac{H-1}{H}}T(A_H+B_Hu) \Bigr) \leq C_H \exp(-u^{2H}).$$

This "stretched exponential" decay for deviations implies the following moment-generating behavior:

• For $H > 1/2$, $2H > 1$, hence the decay is faster than pure exponential, and $\mathbb{E}[\exp(\alpha\, TV^\delta(B^H))] < \infty$ for all $\alpha > 0$.
• For $H = 1/2$, corresponding to standard Brownian motion, the tail decays like $\exp(-u)$, so the exponential moment exists only below a critical threshold of $\alpha$.
• For $H < 1/2$, $2H < 1$, so the decay is not integrable at infinity, and $\mathbb{E}[\exp(\alpha\, TV^\delta(B^H))] = \infty$ for every $\alpha > 0$ (Bednorz et al., 2012).

3. Concentration Inequalities and Large Deviations

Sharp deviation inequalities for the truncated variation center around the conditional mean $\mu_T(\delta) = \mathbb{E}[TV^\delta(B^H)_0^T]$.

For $H\in(0, \tfrac12)$ (negatively correlated increments), with $\delta \leq T^H$:

$$\Pr\bigl(|TV^\delta(B^H)_0^T - \mu_T(\delta)| > x\bigr) \leq \bar A_H \exp\left( - \bar B_H T^{-1}\delta^{1/H - 2} x^{1+2H} \min\{T\delta^{1-1/H}, x^{1-2H}\} \right)$$

for constants $\bar A_H \leq 36$, $\bar B_H>0$ depending on $H$.

For $H\in[\tfrac12,1)$ (positively correlated increments), with $\delta \leq T^H$:

$$\Pr(|TV^\delta(B^H)_0^T - \mu_T(\delta)| > x) \leq \bar A \exp\left(-\bar B T^{-2H} x^2 \right)$$

with universal constants $\bar A \leq 4$, $\bar B \geq 2/\pi^2$.

These results establish sub-Gaussian concentration (i.e., Gaussian-type tails) for $H\geq 1/2$ and heavier-tailed, stretched-exponential concentration for $H<1/2$ (Bednorz et al., 16 Dec 2025, Bednorz et al., 2012). The bounds are optimal up to constants for large deviation scales.

4. Proof Techniques: Chaining, Gaussian Suprema, and Metric Entropy

The probabilistic analysis of truncated variation leverages a combination of chaining techniques, metric entropy, and Gaussian supremum concentration:

• The truncated variation (and one-sided variants) is represented as a supremum over finitely many affine functions of a centered Gaussian vector.
• Fresen's Gaussian-supremum concentration inequality is employed for finite-dimensional approximations.
• Sharp control requires norming by increment variances, and the analysis differentiates regimes with positive versus negative increment correlation (i.e., $H \ge 1/2$ versus $H < 1/2$).
• Passage to the continuum involves a discretization limit, exploiting the self-similarity and stationary increments of fBm (Bednorz et al., 16 Dec 2025, Bednorz et al., 2012).

Corollary results elucidate the influence of domain length and truncation parameter on concentration and scaling.

5. $1/H$-Variation and Crossing Functional

For $fBm$ on $[0,T]$, the $1/H$-variation along Lebesgue partitions is analyzed in terms of strip (level) crossings. Define

$$K_{s,t}(c, f) := \sum_{p\in\mathbb{Z}} N_{s,t}(c, f - p c),$$

where $N_{s,t}(c, g)$ counts up- or down-crossings of the strip $\{y \in [0,c]\}$. For $c \to 0$,

$$V^{(1/H)}(B^H)_0^T := \lim_{c\to 0} c^{1/H} K_{0,T}(c, B^H) = \mathfrak c_H T \quad \text{a.s.}$$

with constant $\mathfrak c_H > 0$. Tail probabilities for $K_{s,t}(c, B^H-\rho)$ derive from the sharp concentration theory, with stretched exponential decay in $v$, reflecting the underlying roughness of $B^H$ sample paths (Bednorz et al., 16 Dec 2025).

6. Asymptotic Behavior, Mean Scaling, and Law of Large Numbers

Self-similarity and scaling properties of $B^H$ directly yield the asymptotic mean behavior for small truncation:

$$\mathbb{E}\,TV^\delta(B^H;[0,T]) \asymp T\,\delta^{(H-1)/H} \qquad \text{as }\delta\to 0$$

with constants dependent only on $H$. For $H = 1/2$,

$\mathbb{E}\,TV^\delta(B^{1/2}) \sim T/\delta$

and $c\,TV^c(B^{1/2}) \to T$ almost surely as $c\downarrow 0$, reflecting the law-of-large-numbers for partition-wise sum functionals of Brownian motion (Bednorz et al., 2012).

7. Connection to Local Times

The strip crossing counts for small $c$ recover the local time density $L^H(T,a)$ of $B^H$ at time $T$. For a bounded test function $g$,

$$\int_{\mathbb{R}} c^{1/H-1} U_{0,T}(c, B^H - a) g(a) da \longrightarrow \frac{\mathfrak c_H}{2} \int_{\mathbb{R}} L^H(T,a) g(a) da \quad \text{almost surely as } c\to 0,$$

and the random measures

$c^{1/H-1} U_{0,T}(c, B^H - \cdot)\, da$

converge almost surely (weakly in $L^1$) to $(1/2)\mathfrak{c}_H L^H(T, \cdot) da$ (Bednorz et al., 16 Dec 2025). This establishes a direct pathwise link between the truncated variation, crossing counts, and local time structure of fBm.

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These results collectively provide an essentially optimal description of the distributional and almost-sure behavior of the truncated variation functional of fBm, including precise large deviation bounds, moment asymptotics, and ergodic-type scaling limits. The technical framework, utilizing Gaussian concentration, chaining, and partition-supremum methods, is robust for extensions to related Gaussian processes and sample path functionals.