Multivariate Linear Single Big Jump Principle

• The multivariate linear single big jump principle is a framework that explains how rare extreme events in heavy-tailed multivariate aggregates are overwhelmingly driven by a single large jump.
• It establishes asymptotic equivalences for both fixed and random sums, demonstrating that the tail behavior of aggregates is primarily determined by one dominating term.
• The theory underpins efficient simulation and robust risk management strategies in applications such as finance and actuarial science through precise, actionable large deviation results.

The multivariate linear single big jump principle is a rigorous conceptual and quantitative framework that explains how rare events in the tails of sums or aggregates of multivariate, heavy-tailed random variables are asymptotically dominated by a single large “jump”. This principle generalizes the classical univariate big jump phenomenon to linear and multivariate contexts, particularly in jump processes, risk modeling, and applied probability, providing key asymptotic results and operational insights for real-world systems where high thresholds or rare fluctuations are of concern.

1. Formulation and Theoretical Structure

The core formulation of the single big jump principle in the multivariate linear context asserts that, for sums $S_n = X_1 + X_2 + \dots + X_n$ of independent (or weakly dependent) random vectors $X_i$ in $\mathbb{R}^d$:

$$\mathbb{P}(S_n \in xA) \sim \sum_{i=1}^n \mathbb{P}(X_i \in xA), \qquad x \to \infty$$

for suitably chosen rare sets $A \subseteq \mathbb{R}^d$ (typically convex cones or increasing sets), and for sequences $\{X_i\}$ whose distribution belongs to a heavy-tailed class such as multivariate subexponential ($S_A$), dominatedly varying ($D_A$), or strong subexponential ($S_A^*$) distributions (Konstantinides et al., 14 Oct 2024, Passalidis, 28 Mar 2025). This property extends classical results (e.g., for subexponential distributions) by showing that, as the threshold becomes large, the probability that the sum lands in a rare set is governed by the likelihood that any one of the summands alone is exceptional.

For randomly stopped sums $S_T = X_1 + \dots + X_T$ with $T$ an integer-valued random variable independent of the $X_i$, the single big jump principle yields the asymptotic equivalence

$$\mathbb{P}(S_T \in xA) \sim \mathbb{E}[T] \, \mathbb{P}(X \in xA), \qquad x \to \infty,$$

under the condition that the tail of $T$ is lighter than that of $X$ (Passalidis, 28 Mar 2025).

The principle further entails that in the conditional distribution, given a rare event (e.g., $S_n \in xA$), exactly one (or, under truncation, the minimal number) of the $X_i$ takes on an exceptionally large value; all others are typical (Kerriou et al., 2022). In certain models, if componentwise truncation exists, a generalized "fewest-big-jumps" variant arises, identifying the minimal number $k$ of summands required to realize the excess (Kerriou et al., 2022).

In infinitely divisible multivariate models, the tail of the distribution is characterized in terms of the normalized Lévy measure $V_{A,1}(x)$, and the equivalence

$F_A(x) \sim V_{A,1}(x)$

connects the single big jump effect to the structure of the underlying process (Konstantinides et al., 14 Oct 2024).

2. Closure Properties and Distributional Classes

Closure properties for multivariate single big jump classes are systematically established in recent works. The primary classes—multivariate long-tailed ($L_A$), dominatedly varying ($D_A$), subexponential ($S_A$), and strong subexponential ($S_A^*$)—are examined under the following operations (Konstantinides et al., 14 Oct 2024, Passalidis, 28 Mar 2025):

• Product convolution: If $X \in D_A$ and $\Omega$ is an independent random vector, then $\Omega X$ preserves the $D_A$ property.
• Scale mixture: For proportional scaling, if $X \in L_A$, then the mixture remains in $L_A$.
• Convolution/Finite Mixtures: For $X_1$, $X_2$ independent with $X_i \in D_A$ (or related classes), $X_1 + X_2$ yields

  $F_1 * F_2(xA) \sim F_1(xA) + F_2(xA)$

  for large $x$.

These closure results justify the robustness of the principle under aggregation, mixing, and transformation, which are integral operations in multivariate modeling, actuarial science, and dependent risk processes.

3. Precise Large Deviations and Asymptotic Results

Recent work extends the single big jump principle to deliver uniform and precise large deviation asymptotics for both random and deterministic sums. For $X(i) \in S^*_A$ (strong subexponential), for any fixed $n$,

$$\mathbb{P}\left(S_n \in xA\right) \sim n \, \mathbb{P}(X \in xA), \qquad x \to \infty,$$

with similar results holding for randomly indexed sums under appropriate counting process conditions (Passalidis, 28 Mar 2025). Importantly, these results are uniform for $x$ above a sequence defined via an insensitivity function $h$ reflecting slow variation properties of the tail.

In risk models with discounting and investment, such as the present value of claims

$D(T) = \sum_{i=1}^{N(T)} X(i) e^{-R_{T_i}},$

with $N(T)$ the claim count process and $R_{T_i}$ a stochastic discount factor, the big jump principle gives (Konstantinides et al., 14 Oct 2024, Konstantinides et al., 20 Oct 2025):

$$\mathbb{P}(D(T) \in xA) \sim \int_0^T \mathbb{P}(X e^{-R_t} \in xA) dt, \qquad x \to \infty.$$

If claims and financial risks are weakly or arbitrarily dependent, or governed by a Lévy process, refinement and explicit asymptotics are available under minimal regularity assumptions.

4. Applications in Risk Models and Aggregate Claims

The multivariate linear single big jump principle has direct implications in actuarial mathematics, finance, queueing networks, and other high-dimensional risk systems. In comprehensive risk models (Konstantinides et al., 14 Oct 2024, Konstantinides et al., 20 Oct 2025):

• Discounted Claim Sums: For d-dimensional claim process with common Poisson/renewal arrivals and a financial risk process, the tail of discounted aggregate claims or the infinite-horizon ruin probability is asymptotically equivalent to the sum or integral over individual claim events, each discounted, reflecting the "one claim dominates" phenomenon even in the multivariate setting.
• Dependence Structure: The main theorems cover both weak and arbitrary dependence between the insurance portfolio and financial market returns.
• Explicit Formulas: In multivariate regularly varying (MRV) regimes, tail probabilities can be written explicitly in terms of tail measures, auxiliary distribution functions (e.g., $G(x) \sim x^{-\alpha} L(x)$), and expectations over Lévy exponents.

This understanding streamlines actuarial computation, supports tangible risk capital allocation, and suggests that advanced methods (e.g., Monte Carlo, simulation) should emphasize the accurate sampling of rare, single large jump events.

5. Operational and Practical Implications

The theory facilitates:

• Efficient Simulation: Targeting only the dominant large claim(s) for rare event simulation improves estimator efficiency, as shown in unbiased Monte Carlo methods (Chen et al., 2021).
• Risk Management: Capital and reserve calculations focusing on single big loss events provide conservative and realistic risk assessments.
• Extreme Event Prediction: In application domains from environmental risk to network congestion, the frequency and scale of catastrophic events are directly computable in terms of the single big jump asymptotics.

Moreover, the big jump principle remains stable under practically relevant data transformations, such as random scaling (modeling investment risk) or convolution (portfolio aggregation).

6. Comparative Analysis and Recent Advances

Compared with earlier literature, these recent works (Konstantinides et al., 14 Oct 2024, Passalidis, 28 Mar 2025, Konstantinides et al., 20 Oct 2025) generalize big jump phenomena to more inclusive multivariate classes, prove closure under product convolution and scale mixtures, and relax conditions (e.g., on the counting process tail) to the minimal necessary. The asymptotic formulas hold for both random and deterministic stopping times and provide sharper characterization than prior approaches restricted to the MRV or univariate subexponential settings.

Furthermore, the methodology demonstrates that, even with multivariate dependencies, discounting, or non-renewal arrival processes, the probability of extreme aggregate outcomes continues to be asymptotically governed by a single dominating component or claim.

7. Summary Table of Key Results

Setting	Asymptotic Tail Probability	Comment
Sum of n i.i.d. vectors in $S_A$	$\mathbb{P}(S_n \in xA) \sim n \mathbb{P}(X \in xA)$	Deterministic aggregate
Random sum $S_T$ ($T$ independent)	$$\mathbb{P}(S_T \in xA) \sim \mathbb{E}[T] \mathbb{P}(X \in xA)$$	Requires light tail of $T$
Discounted claims, aggregate $D(T)$	$$\mathbb{P}(D(T) \in xA) \sim \int_0^T \mathbb{P}(X e^{-R_t} \in xA) dt$$	With financial factors/Lévy processes
Finite mixture, e.g., $\Omega X$	Tail equivalence with $X$ under $D_A$ or $S_A$	Product convolution, scale mixture

Conclusion

The multivariate linear single big jump principle provides a unifying, rigorously justified explanation for rare-event behavior in sums and aggregates of multivariate heavy-tailed random variables. By establishing that high-threshold events are overwhelmingly due to a single dominating term, the principle extends classical asymptotic results, ensures closure under central probabilistic operations, and supports accurate risk quantification in high-dimensional systems, especially with applications in insurance, finance, and dependent random systems (Konstantinides et al., 14 Oct 2024, Passalidis, 28 Mar 2025, Konstantinides et al., 20 Oct 2025).