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 |
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Keywords
- lamperti’s problem
- almost-sure bounds
- law of large numbers
- central limit theorem
- birth-and-death chain
- transience
- inhomogeneous random walk
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Rate of escape and central limit theorem for the supercritical Lamperti problem. / Menshikov, Mikhail V.; Wade, A.R.
In: Stochastic Processes and their Applications, Vol. 120, No. 10, 09.2010, p. 2078-2099.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 -