Single-Big-Jump Principle

• The Single-Big-Jump Principle is a concept in large deviation theory, showing that extreme aggregate outcomes are primarily due to one dominant event.
• It extends to complex models like Lévy walks and the Lévy–Lorentz gas, capturing rare-event behavior in heavy-tailed and subexponential systems.
• The principle aids in distinguishing regimes where individual extremes govern probabilities, emphasizing its practical implications in stochastic modeling.

A principle fundamental to the theory of large deviations in heavy-tailed and subexponential systems, the Single-Big-Jump Principle asserts that rare, extreme fluctuations of an aggregate quantity are overwhelmingly caused by a single dominant event (or "jump") rather than by the collective effect of many moderate fluctuations. Rigorous in the context of independent and identically distributed (IID) subexponential summands, the principle has been extended to encompass Lévy walks, quenched-disorder models such as the Lévy–Lorentz gas, and continuous-time random walks with heavy-tailed and stretched exponential increment distributions. The principle’s reach and limitations are central to rare-event analysis, non-Gaussian extreme statistics, and the understanding of rare-event mechanisms in complex stochastic processes (Burioni et al., 2019).

1. Rigorous Statement and Subexponential Foundations

Consider non-negative IID random variables $X_1,\dots,X_n$ with distribution $F$ and right tail $\overline{F}(x)=P(X>x)$. Subexponentiality is defined by

$$\lim_{x \to \infty} \frac{P(X_1 + \cdots + X_n > x)}{\overline{F}(x)} = n \quad \forall n \geq 2.$$

This is equivalent to the property: $$\lim_{x\to\infty}\frac{P(X_1+X_2>x)}{\overline F(x)} = 2$$ for two summands. The main theorem states: $$P\left( S_n > x \right) \sim n \overline F(x), \qquad x \to \infty,\quad S_n = \sum_{i=1}^n X_i$$ where $\overline{F}(x)$ is subexponential. The implication is that for rare-event exceedances by large $x$, the probability is dominated by paths in which a single $X_i$ is of size $x$—a "single-big-jump" dominates, rather than an accumulation of jointly moderate deviations (Burioni et al., 2019).

This mechanism is characteristic of regularly varying (power-law) and subexponential distributions, where the decay of $\overline{F}(x)$ is slow compared to any exponential.

2. Extensions to Lévy Walks and Lévy–Lorentz Gas

The principle generalizes to more complex stochastic processes, notably Lévy walks (step lengths or waiting times drawn from broad-tailed distributions) and models with quenched disorder like the Lévy–Lorentz gas. For a Lévy walk, step durations are IID with heavy tail: $$\lambda(x) \sim C x^{-1-\mu},\quad 0<\mu<2,\quad x \to \infty.$$ The rare-event tail of displacement $r$ at time $t$ is captured by the ansatz: $$B(r,t) \approx \int_0^t dt_w \int_0^\infty dx\; r_N(t_w)\, \lambda(x)\; \mathcal{P}(r|t,x,t_w),$$ where $r_N(t_w)$ is the rate of initiation of steps by $t_w$ and $\mathcal{P}(r|t,x,t_w)$ is the conditional probability that a step of size $x$ at $t_w$ gives displacement $r$ at $t$. The same effective-rate formalism applies to the Lévy–Lorentz gas (random gaps between scatterers drawn IID from broad tails), with $r_N(t_w) \propto (t_w\,\tau_0)^{-1/2}$ at large $t_w$ for finite mean gap, and the tail constructed as a sum over big-gap realizations and numbers of reflections (Burioni et al., 2019).

3. Stretched Exponential Distributions and Anomalous Scaling

When jump or gap distributions are stretched exponential (Weibull),

$$\lambda_\alpha(x) = \frac{\alpha}{\tilde{x}} \left( \frac{x}{\tilde{x}} \right)^{\alpha-1} \exp\left[-(x/\tilde{x})^\alpha\right], \quad 0<\alpha<1,$$

the single-big-jump principle still applies but introduces a nontrivial second scale ($\tilde{x}$) alongside the ballistic or diffusive scale ($vt$ for fixed speed $v$). The rare-event PDF tail exhibits the form: $$B(r,t) = \frac{1}{v\langle t \rangle} \exp\left[-(r/(v\tilde{t}))^\alpha\right] \left[\,1 + \alpha (r/(v\tilde{t}))^\alpha \left( \frac{v t}{r} - 1 \right) \right]$$ with summing over reflection paths in the Lorentz gas. For $\alpha<1$, moments remain finite and show diffusive scaling $\langle r^q(t)\rangle\sim t^{q/2}$, but the non-Gaussian rare-event structure survives. For $\alpha>1$, the single-big-jump approximation fails and standard large-deviation estimates recover dominance (Burioni et al., 2019).

4. Mechanistic Consequence and Boundary of Applicability

The principle holds under the assumption that the rare event (e.g., $S_n > x$) can occur via a single summand of size $x$, and is valid for all subexponential cases ($\alpha<1$ for Weibull). If the tail becomes thin ($\alpha>1$), or in certain correlated or functional settings, multiple large jumps or joint deviations are required, invalidating the naive single-jump approximation. Then, rare-event probabilities scale as higher-order products, requiring “multiple-jump” analysis wherein the minimal number $l^*$ of big jumps must be identified, and probabilities scale as $\sim [n \overline{F}(x)]^{l^*}$ up to constants (Chen et al., 2017).

5. Effective-Rate Ansatz and Path Decomposition

A methodological core is the effective-rate formalism for computing rare-event tails in non-IID settings. The probability of a rare fluctuation is factorized into the rate of attempts to realize a big jump and the conditional outcome of each attempt: $$B(r,t) = \int_0^t dt_w \int_0^\infty dx \; r_N(t_w) \lambda(x) \mathcal{P}(r|t,x, t_w)$$ This incorporates physical constraints (e.g., finite time, possible reflections) and captures nonanalyticities (such as at rational $r/(vt)$ in random Lorentz gases), which cannot be resolved by conventional central-limit or standard large deviation methods (Burioni et al., 2019).

6. Singular Features, Scaling, and Extensions

The analytic forms provided for rare-event PDF tails display non-Gaussian behavior, the survival of rare paths due to a single exceptional increment, and, for specific models, singularities and cusps in scaling functions. The construction captures the physical and probabilistic fingerprints of subexponential mechanisms—such as the dominance of the maximal excursion, the dynamical emergence of non-universal scaling, and the precise mechanism of the rare event. These features are robust as long as independence (or weak correlations) and subexponential tails persist. Breakdown of these assumptions necessitates more general frameworks (renewal analysis, multiple-jump large deviations, pathwise large deviation theory) (Burioni et al., 2019, Chen et al., 2017).

7. Discussion and Implications

The Single-Big-Jump Principle provides a universal predictive tool for rare-event estimation in systems with heavy-tailed or subexponential fluctuations. Its validity demarcates the boundary between regimes controlled by rare individual extremes and those requiring organized collective deviations. The principle’s reach into complex dynamical models substantiates its foundational role in rare-event analysis, but also identifies scenarios—thin tails, strongly correlated increments, or multi-jump events—where it ceases to dominate. Extensions to such regimes require developed large deviation theory for multiple-jump or functional events, renewing the interplay between probabilistic structure and stochastic modeling (Burioni et al., 2019, Chen et al., 2017).