Drift Inference for Unit-Root Galton-Watson Processes with Immigration
Abstract
We study inference on the drift of a critical Galton--Watson process with immigration, a count time series with a unit root. Climate change motivates such nonstationary models for weather-related disaster counts. The drift is the expected increase in the count per period. Ordinary least squares is inconsistent for the drift, so we study state-weighted least squares, giving greater weight to observations following small counts, where conditional variance is lower. Under strict recurrence, we apply null-recurrent regenerative limit theory to obtain a polynomial convergence rate and a standard normal studentized limit. Our main contribution is the recurrence boundary, where we establish a logarithmic convergence rate and a parameter-free non-Gaussian studentized limit. Estimating the optimal weights yields the same first-order limiting distribution as knowing them. Simulations show lower root mean squared error than time-weighted least squares with weights 1/t, and improved state-weighted confidence-interval coverage when the unit root is imposed.
Related
- Asymptotic behavior of some strongly critical decomposable multi-type Galton-Watson processes with immigration
- Functional limit theorems for Galton--Watson processes with inhomogeneous immigration
- Functional limit theorems for Galton-Watson processes with inhomogeneous immigration towards a deterministic limit function
- Biased random walk on a locally infinite Galton-Watson tree
- Infinite-mean Galton-Watson trees have zero random-interchange threshold
BibTeX
@misc{lu2026drift,
title = {{Drift Inference for Unit-Root Galton-Watson Processes with Immigration}},
author = {Yang Lu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.17999},
eprint = {2609.17999},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17999},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Drift Inference for Unit-Root Galton-Watson Processes with Immigration
AU - Yang Lu
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2609.17999
UR - https://arxiv.org/abs/2609.17999
AB - We study inference on the drift of a critical Galton--Watson process with immigration, a count time series with a unit root. Climate change motivates such nonstationary models for weather-related disaster counts. The drift is the expected increase in the count per period. Ordinary least squares is inconsistent for the drift, so we study state-weighted least squares, giving greater weight to observations following small counts, where conditional variance is lower. Under strict recurrence, we apply null-recurrent regenerative limit theory to obtain a polynomial convergence rate and a standard normal studentized limit. Our main contribution is the recurrence boundary, where we establish a logarithmic convergence rate and a parameter-free non-Gaussian studentized limit. Estimating the optimal weights yields the same first-order limiting distribution as knowing them. Simulations show lower root mean squared error than time-weighted least squares with weights 1/t, and improved state-weighted confidence-interval coverage when the unit root is imposed.
ER -