Beyond the Big Jump: A Perturbative Approach to Stretched-Exponential Processes

Abstract

The problem of sums of independent, identically distributed random variables with stretched-exponential tails exhibits a dynamical phase transition and has recently reemerged in the context of active transport and condensation phenomena. We develop a perturbative expansion for the distribution of the sum that systematically extends the Big Jump Principle beyond its asymptotic regime. The expansion yields explicit higher order corrections that describe moderate deviations, bridging the gap between typical Gaussian fluctuations and the far-tail behavior dominated by single big jump events. In this sense, our approach is complementary to the classical Edgeworth expansion, which provides corrections to the Gaussian core, whereas we construct systematic corrections to the big jump regime. The leading terms reveal the scaling structure governing the crossover between typical and condensed fluctuations, in agreement with large deviation predictions but without relying on its asymptotic limit. We further extend the framework to continuous-time random walks (CTRWs), where stretched-exponential jump statistics combined with stochastic renewal times generate nontrivial propagators through subordination. This setting is particularly relevant for transport processes with non-Gaussian displacement statistics, where super-exponential or Laplace-like tails emerge from the interplay of rare large jumps and temporal fluctuations. All analytical predictions are supported by numerical simulations.

Figures (6)

• Figure 1: Graphical representation of $g(\vec{y})$ for $n=3$ jumps (red surface), with an overlay of $g(\vec{y})$ for $n=2$ (blue line), shown for increasing values of $\beta = \{0.3,\,0.5,\,0.9\}$. The function exhibits one central cusp at $\vec{y} = 0$ and $n-1$ outskirt cusps located along the integration axes, each corresponding to a distinct realization of the Big Jump event occurring in one of the summands. As $\beta \to 1^{-}$, the stretched-exponential distribution approaches a piecewise-linear form, and the cusps become more linear; the convergence to the leading BJP term \ref{['phi_x_n_BJP']} correspondingly slows down.
• Figure 2: Plots of our perturbative correction obtained in Eq.\ref{['phi_general_n_G0']} (dashed colored lines) compared to the numerical results (black crosses), for $\beta = 0.4$ (left), $\beta = 0.5$ (middle), and $\beta = 0.9$ (right). The number of jumps is set to $n=30$, and the plots are in log-scale and normalized with respect to the BJP asymptotic limit $nf(x)$. While for any $\beta<1$ and $x\to\infty$ the BJP leading term converges to the numerical results, for finite values of $x$ and incresasing $\beta$ the higher order correction terms becomes more relevant. The labels $L$ means a $L$-th truncated perturbative series, i.e. $G^{(L)}(x)$. In these plots we use $L=2, 4, 8, 20$.
• Figure 3: Location of the minimal term $\mathcal{G}^{(L^*)}(x)$ of the perturbative series $G(x)$ as a function of $x$ and $\beta$. As $x$ increases, a larger number of perturbative terms remain non-divergent, and the optimal truncation order $L^*(x)$ grows. For moderate $x$, only a few correction terms should be retained. As $\beta\to1^{-}$, convergence to the BJP becomes slower and more terms are required. The plots are shown in log-scale for $x=5, 10, 20, 30$, with $\beta=0.4$ (left), $\beta=0.5$ (middle) and $\beta=0.9$ (right).
• Figure 4: Conditional PDF $\phi(x|n)$ normalized by the BJP asymptotic behavior $n f(x)$ (black crosses). Dashed colored curves show perturbative approximations from Eq. \ref{['phi_general_n_G0']} truncated at orders $L=2$ and $L=20$. The orange line shows the CLT prediction from Eq. \ref{['phi_x_n_clt']}. Red circles represent the optimally truncated theory using $G^{(L^*)}(x)$. Simulations are shown in log-scale for $n=30$, with $\beta=0.4$ (left), $\beta=0.5$ (middle) and $\beta=0.9$ (right). The optimal truncation provides excellent agreement with numerical results even for moderate $x$.
• Figure 5: Numerical simulations of the sum of IID stretched-exponential random variables distributed according to Eq. \ref{['f_x']}. Results are shown for $\beta=0.5$ and two values of the number of summands, $n=7$ and $n=100$. Blue crosses represent the empirical rate function extracted from simulations. The yellow parabola shows the quadratic rate function associated with typical Gaussian fluctuations predicted by the CLT. The black curve shows the full rate function given by Eq. \ref{['I_exact']}, obtained from large-deviation theory in the limit $n\to\infty$. The green dotted curve shows the leading BJP contribution $\alpha^{\beta}|r|^{\beta}$, while the green dashed curve shows the large $r$ approximation $\mathcal{I}_{approx}(r)$ from Eq. \ref{['I_approx']} derived from the perturbative expansion. For moderate $n$, the numerical data interpolate smoothly between the Gaussian regime and the large $r$ behaviour. The perturbative large $r$ approximation captures the behaviour of the distribution in the big jump regime and provides a good description of moderate deviations, even away from the strict large deviation limit.