New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach

Laulin, Lucile

doi:10.1007/s10955-021-02834-x

Cited by 6 publications

(11 citation statements)

References 23 publications

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“…Proof of Theorem 2.2. We need to check that all the hypotheses of Theorem A.3 in [19] are satisfied. Thanks to Lemma 4.1, hypothesis (H.1) holds almost surely.…”

Section: The Diffusive Regimementioning

confidence: 99%

“…Thus, (4.16) guarentees that (H.4) is verified. We are now going to apply the quadratic strong law given by Theorem A.3 in [19]. We get from equation (4.15) that…”

Section: The Diffusive Regimementioning

confidence: 99%

“…The ERW with stops or minimal RW, changing in particular the distribution of α n , has also been investigated [5,4,14,20]. The ERW with reinforced memory has been studied by Baur [1] via the urn approach, and Laulin [19] using martingales.…”

Section: Introductionmentioning

confidence: 99%

“…The idea of this paper is to use the approach developped in [7] and [19] to study how changing the memory allows us to induce amnesia to the ERW. More precisely, the distribution of the memory β n of our new variation of the ERW is such that the probability of choosing a fixed instant k ∈ N * at time n ≥ k decreases approximatly with speed (k/n) β for some amnesia parameter β ≥ 0.…”

Section: Introductionmentioning

confidence: 99%

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Introducing smooth amnesia to the memory of the Elephant Random Walk

Laulin¹

2022

Preprint

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This paper is devoted to the asymptotic analysis of the amnesic elephant random walk (AERW) using a martingale approach. More precisely, our analysis relies on asymptotic results for multidimensional martingales with matrix normalization. In the diffusive and critical regimes, we establish the almost sure convergence and the quadratic strong law for the position of the AERW. The law of iterated logarithm is given in the critical regime. The distributional convergences of the AERW to Gaussian processes are also provided. In the superdiffusive regime, we prove the distributional convergence as well as the mean square convergence of the AERW.

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“…Proof of Theorem 2.2. We need to check that all the hypotheses of Theorem A.3 in [19] are satisfied. Thanks to Lemma 4.1, hypothesis (H.1) holds almost surely.…”

Section: The Diffusive Regimementioning

confidence: 99%

“…Thus, (4.16) guarentees that (H.4) is verified. We are now going to apply the quadratic strong law given by Theorem A.3 in [19]. We get from equation (4.15) that…”

Section: The Diffusive Regimementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Introducing smooth amnesia to the memory of the Elephant Random Walk

Laulin¹

2022

Preprint

Self Cite

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show abstract

“…The ERW with stops or minimal RW, changing in particular the distribution of α n , has also been investigated [4,5,12,18]. The ERW with reinforced memory has been studied by Baur [2] via the urn approach, and Laulin [17] using martingales.…”

Section: Introductionmentioning

confidence: 99%

Introducing smooth amnesia to the memory of the Elephant Random Walk

Laulin

2022

Electron. Commun. Probab.

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Broadcasting‐induced colorings of preferential attachment trees

Desmarais

Holmgren

Wagner

2023

Random Struct Algorithms

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We consider random two‐colorings of random linear preferential attachment trees, which includes recursive trees, plane‐oriented recursive trees, binary search trees, and a class of d‐ary trees. The random coloring is defined by assigning the root the color red or blue with equal probability, and all other vertices are assigned the color of their parent with probability p$$ p $$ and the other color otherwise. These colorings have been previously studied in other contexts, including Ising models and broadcasting, and can be considered as generalizations of bond percolation. With the help of Pólya urns, we prove limiting distributions, after proper rescalings, for the number of vertices, monochromatic subtrees, and leaves of each color, as well as the number of fringe subtrees with two‐colorings. Using methods from analytic combinatorics, we also provide precise descriptions of the limiting distribution after proper rescaling of the size of the root cluster; the largest monochromatic subtree containing the root.

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New Insights on the Reinforced Elephant Random Walk Using a Martingale Approach

Cited by 6 publications

References 23 publications

Introducing smooth amnesia to the memory of the Elephant Random Walk

Introducing smooth amnesia to the memory of the Elephant Random Walk

Introducing smooth amnesia to the memory of the Elephant Random Walk

Broadcasting‐induced colorings of preferential attachment trees

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