## The Annals of Applied Probability

### Thresholds for detecting an anomalous path from noisy environments

#### Abstract

We consider the “searching for a trail in a maze” composite hypothesis testing problem, in which one attempts to detect an anomalous directed path in a lattice 2D box of side $n$ based on observations on the nodes of the box. Under the signal hypothesis, one observes independent Gaussian variables of unit variance at all nodes, with zero mean off the anomalous path and mean $\mu_{n}$ on it. Under the null hypothesis, one observes i.i.d. standard Gaussians on all nodes. Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] showed that if the unknown directed path under the signal hypothesis has known initial location, then detection is possible (in the minimax sense) if $\mu_{n}\gg1/\sqrt{\log n}$, while it is not possible if $\mu_{n}\ll1/\log n\sqrt{\log\log n}$. In this paper, we show that this result continues to hold even when the initial location of the unknown path is not known. As is the case with Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757], the upper bound here also applies when the path is undirected. The improvement is achieved by replacing the linear detection statistic used in Arias-Castro et al. [Ann. Statist. 36 (2008) 1726–1757] with a polynomial statistic, which is obtained by employing a multiscale analysis on a quadratic statistic to bootstrap its performance. Our analysis is motivated by ideas developed in the context of the analysis of random polymers in Lacoin [Comm. Math. Phys. 294 (2010) 471–503].

#### Article information

Source
Ann. Appl. Probab., Volume 28, Number 5 (2018), 2635-2663.

Dates
Revised: September 2017
First available in Project Euclid: 28 August 2018

https://projecteuclid.org/euclid.aoap/1535443230

Digital Object Identifier
doi:10.1214/17-AAP1356

Mathematical Reviews number (MathSciNet)
MR3847969

Zentralblatt MATH identifier
06974761

#### Citation

Chatterjee, Shirshendu; Zeitouni, Ofer. Thresholds for detecting an anomalous path from noisy environments. Ann. Appl. Probab. 28 (2018), no. 5, 2635--2663. doi:10.1214/17-AAP1356. https://projecteuclid.org/euclid.aoap/1535443230

#### References

• [1] Arias-Castro, E., Candès, E. J., Helgason, H. and Zeitouni, O. (2008). Searching for a trail of evidence in a maze. Ann. Statist. 36 1726–1757.
• [2] Berger, Q. and Lacoin, H. (2017). The high-temperature behavior for the directed polymer in dimension $1+2$. Ann. Inst. Henri Poincaré Probab. Stat. 53 430–450.
• [3] Gamkrelidze, N. G. and Rotar’, V. I. (1978). The rate of convergence in a limit theorem for quadratic forms. Theory Probab. Appl. 22 394–397.
• [4] Götze, F. and Tikhomirov, A. N. (1999). Asymptotic distribution of quadratic forms. Ann. Probab. 27 1072–1098.
• [5] Lacoin, H. (2010). New bounds for the free energy of directed polymers in dimension $1+1$ and $1+2$. Comm. Math. Phys. 294 471–503.
• [6] Rotar’, V. I. (1974). Certain limit theorems for polynomials of degree two. Theory Probab. Appl. 18 499–507.
• [7] Rotar’, V. I. and Shervashidze, T. L. (1986). Some estimates for distributions of quadratic forms. Theory Probab. Appl. 30 585–590.
• [8] Whittle, P. (1960). Bounds for the moments of linear and quadratic forms in independent variables. Theory Probab. Appl. 5 302–305.