论文标题
固定高斯序列加权总和的持久性概率
Persistence probabilities of weighted sums of stationary Gaussian sequences
论文作者
论文摘要
以$ \ {ξ_i\} _ {i \ ge 0} $为中心的固定高斯序列，具有非负相关函数$ρ（i）：= \ m athbb {e} [E} [E} [eCy_0Epce_i]加权总和$ \ sum_ {i = 1}^\ellσ（i）ξ_i$，$ \ ell \ ge 1 $的持久概率。对于可总结的相关性$ρ$，我们表明持久性指数是通用的。相反，对于不可能的$ρ$，即使对于多项式权重函数$σ（i）\ sim i^p $，持久性指数取决于相关的衰减速率（由参数$ h $编码）和多项式利率$ p $ $ p $ $σ$。在这种情况下，我们显示了持久性指数$θ（h，p）$的存在，并研究其属性作为$（p，h）$的函数。在证明过程中，我们开发了几种工具来处理具有非负相关性的高斯流程的退出问题 - 例如，持续指数的连续性结果以及持久性指数为零的必要和足够标准为零 - 可能是独立的利益。
With $\{ξ_i\}_{i\ge 0}$ being a centered stationary Gaussian sequence with non-negative correlation function $ρ(i):=\mathbb{E}[ ξ_0ξ_i]$ and $\{σ(i)\}_{i\ge 1}$ a sequence of positive reals, we study the asymptotics of the persistence probability of the weighted sum $\sum_{i=1}^\ell σ(i) ξ_i$, $\ell\ge 1$. For summable correlations $ρ$, we show that the persistence exponent is universal. On the contrary, for non-summable $ρ$, even for polynomial weight functions $σ(i)\sim i^p$ 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.
