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Anomalous fluctuations of renewal-reward processes with heavy-tailed distributions
Statistical Mechanics · Submitted 16 Mar 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3.
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The abstract, sentence by sentence
For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have reveals that this anomalous scaling causes a singularity in the corresponding large deviation function. In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3. For a counting process with the power-law exponent smaller than -3, this anomalous scaling does not take place: this indicates that the processes only fluctuate around the expectation with an error that is compatible with a standard large deviation scaling. In this case, we argue that anomalous scaling appears in higher order cumulants. Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is -3.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. | background 0.98 | 0.14 | 0.11 | asserted 1.00 |
| s1 | Previous works have reveals that this anomalous scaling causes a singularity in the corresponding large deviation function. | background 1.00 | 0.03 | 0.04 | asserted 1.00 |
| s2 | In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). | contribution 0.96 | 0.78 | 0.04 | asserted 0.85 |
| s3 | Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3. | result 1.00 | 0.96 | 0.42 | asserted 1.00 |
| s4 | For a counting process with the power-law exponent smaller than -3, this anomalous scaling does not take place: this indicates that the processes only fluctuate around the expectation with an error that is compatible with a standard large deviation scaling. | result 0.79 | 0.85 | 0.24 | hedged possibility 0.65 |
| s5 | In this case, we argue that anomalous scaling appears in higher order cumulants. | interpretation 0.99 | 0.60 | 0.15 | speculative 0.54 |
| s6 | Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. | method 0.52 | 0.90 | 0.03 | asserted 0.81 |
| s7 | We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is -3. | result 1.00 | 0.94 | 0.15 | asserted 1.00 |
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Original abstract
For renewal-reward processes with a power-law decaying waiting time distribution, anomalously large probabilities are assigned to atypical values of the asymptotic processes. Previous works have reveals that this anomalous scaling causes a singularity in the corresponding large deviation function. In order to further understand this problem, we study in this article the scaling of variance in several renewal-reward processes: counting processes with two different power-law decaying waiting time distributions and a Knudsen gas (a heat conduction model). Through analytical and numerical analyses of these models, we find that the variances show an anomalous scaling when the exponent of the power law is -3. For a counting process with the power-law exponent smaller than -3, this anomalous scaling does not take place: this indicates that the processes only fluctuate around the expectation with an error that is compatible with a standard large deviation scaling. In this case, we argue that anomalous scaling appears in higher order cumulants. Finally, many-body particles interacting through soft-core interactions with the boundary conditions employed in the Knudsen gas are studied using numerical simulations. We observe that the variance scaling becomes normal even though the power-law exponent in the boundary conditions is -3.
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