Persistence of one-dimensional AR(1)-sequences

Persistence probabilities of two-sided (integrated) sums of correlated stationary Gaussian sequences

We study the persistence probability for some two-sided discrete-time Gaussian sequences that are discrete-time analogs of fractional Brownian motion and integrated fractional Brownian motion, respectively. Our results extend the corresponding ones in continuous-time in [Molchan, Commun. Math. Phys., 1999] and [Molchen, J. Stat. Phys., 2017] to a wide class of discrete-time processes.

Persistence probabilities of weighted sums of stationary Gaussian sequences

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 poly…

具有均匀新息的MA(1)序列的持续性概率

Persistence probabilities for MA(1) sequences with uniform innovations

本文研究均匀新息下MA(1)过程的持续性概率，按分布位置与耦合参数划分区域，并在所有区域内显式求出生成函数，部分区域可借助组合量显式表达。

一阶移动平均序列的持久概率及其Laplace创新与q-变形zigzag数

Persistence probabilities of MA(1) sequences with Laplace innovations and $q$-deformed zigzag numbers

本文研究了一阶移动平均序列在Laplace创新下的持久概率，利用q-变形的组合量和生成函数进行显式计算。

Persistence of autoregressive sequences with logarithmic tails

We consider autoregressive sequences $X_n=aX_{n-1}+ξ_n$ and $M_n=\max\{aM_{n-1},ξ_n\}$ with a constant $a\in(0,1)$ and with positive, independent and identically distributed innovations $\{ξ_k\}$. It is known that if $\mathbf P(ξ_1>x)\sim\frac{d}{\log x}$ with some $d\in(0,-\log a)$ then the chains $\{X_n\}$ and $\{M_n\}$ are null recurrent. We investigate the tail behaviour of recurrence times in this case of logarithmically decaying tails. More precisely, we show that the tails of recurrence times are regularly varying of index $-1-d/\log a$. We also prove limit theorems for $\{X_n\}$ and $\…

Universal Persistence for Local Time of One-dimensional Random Walk

We prove the power law decay $p(t,x) \sim t^{-ϕ(x,b)/2}$ in which $p(t,x)$ is the probability that the fraction of time up to $t$ in which a random walk $S$ of i.i.d. zero-mean increments taking finitely many values, is non-negative, exceeds $x$ throughout $s \in [1,t]$. Here $ϕ(x,b)= \mathbb{P}(\text{Lévy}(1/2,κ(x,b))<0)$ for $κ(x,b) = \frac{\sqrt{1-x} b - \sqrt{1+x}}{\sqrt{1-x} b + \sqrt{1+x}}$ and $b=b_S \geq 0$ measuring the asymptotic asymmetry between positive and negative excursions of the walk (with $b_s=1$ for symmetric increments).