arXiv subjects

Shuyuan Liang

Publications and source records attributed to Shuyuan Liang.

5 recordsLinked to original sources

arXiv2026

Precise large deviations for the elephant random walk with Weibull-type and lognormal-type random step sizes

We prove precise large deviations for the symmetric elephant random walk with random step sizes. When the distribution of $|Z_1-1|$ is subexponential, we first show that, for each fixed$p\in[0,1]$, there exists a positive number sequence $b_n\to\infty$ such that \[ \mathbb P(S_n>x)\sim \mathbb P(S_n<-x)\sim \frac n2\overline F(x) \] uniformly for $x\ge b_n$, where $\overline F(x)=\mathbb P(|Z_1-1|>x)$. This existence result requires no moment assumptions and does not specify the growth rate of $b_n$. For $p\in[0,3/4]$, we further give explicit sufficient conditions on $b_n$ under Weibull-type and lognormal-type tail conditions with a regularly varying factor $x^γL(x)$, where $γ\in\mathbf R$ and $L$ is a positive measurable slowly varying function. Analogous results hold with $\mathbb P(|Z_1|>x)$ and non-strict tails.

math.PR↗

arXiv2026

$p$-Wasserstein distance and Cramér-type moderate deviations for the multi-dimensional elephant random walk

In this paper, we establish $p$-Wasserstein bounds and Cramér-type moderate deviations for the multi-dimensional elephant random walk in the diffusive and critical regimes. Using a Gaussian convolution estimate with random covariance and the recursions for the conditional covariance, we obtain $W_{p,2}(\mathcal L(\widehat T_n),N(0,I_d))\le C_{a,d}pL_n$, where $\widehat T_n$ is normalized by its exact covariance and \[ L_n\asymp \begin{cases} n^{-1/2}\log n,&a\le1/3,\\ n^{-3(1/2-a)},&1/3<a<1/2,\\ (\log n)^{-3/2},&a=1/2. \end{cases} \] Consequently, we obtain radial Cramér-type moderate deviations for $x=o(L_n^{-1/3})$. We also determine the covariance bias under the uncorrected normalization and establish the corresponding Wasserstein bounds, lower bounds and moderate deviation estimates. The dimension and the model parameters are fixed.

math.PR↗

arXiv2026

Normal approximation, moderate and large deviations for the elephant random walk with stops and random step sizes

We study the elephant random walk with stops and independent positive random step sizes. Building on fixed-movement estimates, we obtain quantitative normal approximations for an empirically normalized statistic by controlling its random denominator and the independent movement clock. A permutation comparison preserves the diffusive Kolmogorov rate, while variance matching permits third-order Zolotarev estimates. We also derive self-normalized Cramér-type moderate deviations. For regularly varying step-size tails, we obtain uniform precise large deviations and establish the necessity of the diffusive and critical threshold orders. Further applications of the clock representation yield large deviation principles for Weibull-type and light-tailed step sizes, and a precise light-tailed expansion in the balanced case.

math.PR↗

arXiv2026

Normal approximation and self-normalized moderate deviations for the minimal random walk with random step sizes

We study the minimal random walk with independent positive random step sizes. By separating the movement count from the centered step-size noise, we obtain normal approximations and self-normalized Cramér-type moderate deviations under finite-moment assumptions. For the noise, conditioning on the movement count yields Kolmogorov, Wasserstein and Zolotarev bounds. A polynomial moderate-deviation range holds when a recalled stop can lead to movement. At the boundary where this probability is zero, we obtain a logarithmic range and construct a bounded nonsymmetric example excluding a general polynomial range. For the full position, a weighted martingale representation gives limit laws in the different regimes, nonuniform normal approximations and moment-distance bounds without logarithmic losses. Variance concentration also yields self-normalized moderate deviations in the diffusive and critical regimes.

math.PR↗

arXiv2026

Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes

In this paper, we focuses on how the distribution of random step sizes affects Cramér-type moderate deviations and precise large deviations for the elephant random walk with random step sizes. First, with sub-exponential moment condition of random step sizes, we establish Cramér-type moderate deviation results. Consequently, moderate deviation principles are obtained. Secondly, for random step sizes with sub-exponential tails, we establish a large deviation principle on tail probabilities. Finally, for random step sizes with regularly varying tails, we obtain precise large deviations.

math.PR↗