Subleading-order theory for condensation transitions in large deviations of sums of independent and identically distributed random variables

Abstract

We study the full distribution $P_N\left(A\right)$ of sums $A={\sum }_{i=1}^N$ where $x_1,\dots ,x_N$ are $N\gg 1$ independent and identically distributed random variables each sampled from a given distribution $p(x)$ with a subexponential $x\to \infty$ tail. We consider two particular cases: (I) the one-sided stretched exponential distribution $p(x)\propto e^{-x^{\alpha }}$ where $0<x<\infty$, (II) the two-sided stretched exponential distribution $p(x)\propto e^{-|x{|}^{\alpha }}$ where $-\infty <x<\infty$. We assume $0<\alpha <1$ (in both cases). As follows immediately from known theorems, for both cases (i) typical fluctuations of $\Delta A=A-⟨A⟩$ are described by the central-limit theorem, (ii) the tail $A\to \infty$ is described by the big-jump principle $P_N\left(A\right)\simeq Np\left(A\right)$, and (iii) in between these two regimes there is a nontrivial intermediate regime which displays anomalous scaling $P_N\left(A\right)\sim e^{-N^{\beta }f(\Delta A/N^{\gamma })}$ with anomalous exponents $\beta ,\gamma \in (0,1)$ and large-deviation function $f(y)$ that are all exactly known. In practice, although these theoretical predictions of $P_N\left(A\right)$ work very well in regimes (i) and (ii), they often perform quite poorly in the intermediate regime (ii), with errors of several orders of magnitude for $N$ as large as $10^4$. We calculate subleading order corrections to the theoretical predictions in the intermediate regime. We find that for $0<\alpha <{\alpha }_c$, these corrections scale as power laws in $N$, while for ${\alpha }_c<\alpha <1$ they scale as stretched exponentials, where the threshold value is ${\alpha }_c=1/2$ in case (I) and ${\alpha }_c=2/3$ in case (II). This difference between the two cases is a result of the mirror symmetry $p(x)=p(-x)$ which holds only in the latter case.

19 pages, 7 figures

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Cited by in corpus (1)

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