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PreprintCC BY 4.0arXiv

Extremal persistence probabilities of exchangeable sign-invariant random variables

  • Daniel Iľkovič
  • Jun Yan
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Abstract

Let (X1,…,Xn) be a random variable in (ℝ∖{0})n that is both exchangeable and sign-invariant. For every k∈[n], let Sk=∑i=1kXi. Define the weak persistence probability as ℙ(S1,…,Sn≥0), and the strong persistence probability as ℙ(S1,…,Sn>0). From previous results, the optimal lower bound for ℙ(S1,…,Sn≥0) and the optimal upper bound for ℙ(S1,…,Sn>0) are known. We complete the picture by determining the optimal upper bound for ℙ(S1,…,Sn≥0) and the optimal lower bound for ℙ(S1,…,Sn>0) as follows. 1/2nbinomn−1⌊ (n−1)/2⌋≤ℙ(S1,…,Sn>0)≤1/4nbinom2nn≤ℙ(S1,…,Sn≥0)≤1/2nbinomn⌊ n/2⌋. In particular, this implies that the weak and strong persistence probabilities of every exchangeable and sign-invariant random variable (X1,…,Xn) in (ℝ∖{0})n are of the order n−1/2. We also obtain some related results, one in the deterministic setting, and one when the random variables are allowed to take the value 0.

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Cite

BibTeXRIS
BibTeX
@misc{iľkovic2026extremal,
  title = {{Extremal persistence probabilities of exchangeable sign-invariant random variables}},
  author = {Daniel Iľkovič and Jun Yan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05586},
  eprint = {2609.05586},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05586},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Extremal persistence probabilities of exchangeable sign-invariant random variables
AU  - Daniel Iľkovič
AU  - Jun Yan
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05586
UR  - https://arxiv.org/abs/2609.05586
AB  - Let (X_1,…,X_n) be a random variable in (ℝ∖{0})^n that is both exchangeable and sign-invariant. For every k∈[n], let S_k=∑_i=1^kX_i. Define the weak persistence probability as ℙ(S_1,…,S_n≥0), and the strong persistence probability as ℙ(S_1,…,S_n>0). From previous results, the optimal lower bound for ℙ(S_1,…,S_n≥0) and the optimal upper bound for ℙ(S_1,…,S_n>0) are known. We complete the picture by determining the optimal upper bound for ℙ(S_1,…,S_n≥0) and the optimal lower bound for ℙ(S_1,…,S_n>0) as follows. 1/2^nbinomn−1⌊ (n−1)/2⌋≤ℙ(S_1,…,S_n>0)≤1/4^nbinom2nn≤ℙ(S_1,…,S_n≥0)≤1/2^nbinomn⌊ n/2⌋. In particular, this implies that the weak and strong persistence probabilities of every exchangeable and sign-invariant random variable (X_1,…,X_n) in (ℝ∖{0})^n are of the order n^−1/2. We also obtain some related results, one in the deterministic setting, and one when the random variables are allowed to take the value 0.
ER  -