### Abstract

Language | English |
---|---|

Pages | 2078-2099 |

Number of pages | 22 |

Journal | Stochastic Processes and their Applications |

Volume | 120 |

Issue number | 10 |

DOIs | |

Publication status | Published - Sep 2010 |

### Fingerprint

### Keywords

- lamperti’s problem
- almost-sure bounds
- law of large numbers
- central limit theorem
- birth-and-death chain
- transience
- inhomogeneous random walk

### Cite this

*Stochastic Processes and their Applications*,

*120*(10), 2078-2099. https://doi.org/10.1016/j.spa.2010.06.004

}

*Stochastic Processes and their Applications*, vol. 120, no. 10, pp. 2078-2099. https://doi.org/10.1016/j.spa.2010.06.004

**Rate of escape and central limit theorem for the supercritical Lamperti problem.** / Menshikov, Mikhail V.; Wade, A.R.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Rate of escape and central limit theorem for the supercritical Lamperti problem

AU - Menshikov, Mikhail V.

AU - Wade, A.R.

PY - 2010/9

Y1 - 2010/9

N2 - The study of discrete-time stochastic processes on the half-line with mean drift at x given by μ1(x)→0 as x→∞ is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where μ1(x) is of order x−β for some β(0,1). The bounds are of order t1/(1+β), so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where xβμ1(x) has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.

AB - The study of discrete-time stochastic processes on the half-line with mean drift at x given by μ1(x)→0 as x→∞ is known as Lamperti's problem. We give sharp almost-sure bounds for processes of this type in the case where μ1(x) is of order x−β for some β(0,1). The bounds are of order t1/(1+β), so the process is super-diffusive but sub-ballistic (has zero speed). We make minimal assumptions on the moments of the increments of the process (finiteness of (2+2β+ε)-moments for our main results, so fourth moments certainly suffice) and do not assume that the process is time-homogeneous or Markovian. In the case where xβμ1(x) has a finite positive limit, our results imply a strong law of large numbers, which strengthens and generalizes earlier results of Lamperti and Voit. We prove an accompanying central limit theorem, which appears to be new even in the case of a nearest-neighbour random walk, although our result is considerably more general. This answers a question of Lamperti. We also prove transience of the process under weaker conditions than those that we have previously seen in the literature. Most of our results also cover the case where β=0. We illustrate our results with applications to birth-and-death chains and to multi-dimensional non-homogeneous random walks.

KW - lamperti’s problem

KW - almost-sure bounds

KW - law of large numbers

KW - central limit theorem

KW - birth-and-death chain

KW - transience

KW - inhomogeneous random walk

UR - http://www.scopus.com/inward/record.url?scp=77955559322&partnerID=8YFLogxK

UR - http://arxiv.org/abs/0911.2599

U2 - 10.1016/j.spa.2010.06.004

DO - 10.1016/j.spa.2010.06.004

M3 - Article

VL - 120

SP - 2078

EP - 2099

JO - Stochastic Processes and their Applications

T2 - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

IS - 10

ER -