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Persistence probabilities of weighted sums of stationary Gaussian sequences

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  • Aurzada, Frank
  • Mukherjee, Sumit

Abstract

With {ξi}i≥0 being a centered stationary Gaussian sequence with non-negative correlation function ρ(i)≔E[ξ0ξi] and {σ(i)}i≥1 a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum ∑i=1ℓσ(i)ξi, ℓ≥1. For summable correlations ρ, we show that the persistence exponent is universal. On the contrary, for non-summable ρ, even for polynomial weight functions σ(i)∼ip the persistence exponent depends on the rate of decay of the correlations (encoded by a parameter H) and on the polynomial rate p of σ. In this case, we show existence of the persistence exponent θ(H,p) and study its properties as a function of (p,H). During the course of our proofs, we develop several tools for dealing with exit problems for Gaussian processes with non-negative correlations – e.g. a continuity result for persistence exponents and a necessary and sufficient criterion for the persistence exponent to be zero – that might be of independent interest.

Suggested Citation

  • Aurzada, Frank & Mukherjee, Sumit, 2023. "Persistence probabilities of weighted sums of stationary Gaussian sequences," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 286-319.
  • Handle: RePEc:eee:spapps:v:159:y:2023:i:c:p:286-319
    DOI: 10.1016/j.spa.2023.02.003
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    References listed on IDEAS

    as
    1. Lyu, Hanbaek & Sivakoff, David, 2019. "Persistence of sums of correlated increments and clustering in cellular automata," Stochastic Processes and their Applications, Elsevier, vol. 129(4), pages 1132-1152.
    2. Aurzada, Frank & Guillotin-Plantard, Nadine & Pène, Françoise, 2018. "Persistence probabilities for stationary increment processes," Stochastic Processes and their Applications, Elsevier, vol. 128(5), pages 1750-1771.
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