## Chung's law for integrated Brownian motion (1998)

Venue: | Trans. Amer. Math. Soc |

Citations: | 12 - 0 self |

### BibTeX

@ARTICLE{Khoshnevisan98chung'slaw,

author = {Davar Khoshnevisan and Zhan Shi},

title = {Chung's law for integrated Brownian motion},

journal = {Trans. Amer. Math. Soc},

year = {1998},

volume = {350},

pages = {4253--4264}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper is to study the special case of integrated Brownian motion. Let

### Citations

946 |
Continuous martingales and Brownian motion. Third edition. Grundlehren der Mathematischen Wissenschaften 293
- Revuz, Yor
- 1999
(Show Context)
Citation Context ...lies in (0; (t)), which means that the derivative of X vanishes at t , i.e. W (t ) = 0. Therefore in all situations, t 2 [0; (t)] is a zero point of W . According to Corollary VI.2.6 of Revuz and Yor =-=[25]-=-, there exists some s 2 [0; t] such that either t = (s) or t = (s \Gamma ), lim r!s \Gamma (r). In either case, we see that X ((t))ssup 0!st jX((s))j = Y (t). Thus for any given t ? 0, we have proved ... |

633 | Regular variation - Bingham, Goldie, et al. - 1987 |

255 |
Mathematical Analysis of Random Noise
- Rice
- 1954
(Show Context)
Citation Context ...is example (cf. Remark 2.1 below). (b) Integrated Brownian motion is an interesting process on its own. For example, it appears naturally in classical literature in mechanics and biology ([10], [24], =-=[26]-=-; cf. Lachal [19] for a detailed account), as well as in the more recent works in quantum probability ([1], [2], [14]). (c) Our analysis of X uses a genuinely new approach to the small ball problem. W... |

48 |
Metric entropy and the small ball problem for Gaussian measures
- Kuelbs, Li
- 1993
(Show Context)
Citation Context ... an explicit evaluation of (1.1) from the covariance function of X. Even the estimation of the rate of explosion in (1.1) is usually a difficult problem. Indeed, as can be surmised from Kuelbs and Li =-=[16]-=-, this problem is often equivalent to open problems in functional * Research partially supported by grants from the National Science Foundation and the National Security Agency 1 analysis. Known resul... |

29 | Some Aspects of Brownian Motion. Part I. Some special functionals - Yor - 1992 |

26 |
On the maximum partial sums of sequences of independent random variables
- Chung
- 1948
(Show Context)
Citation Context ..., cf. Khoshnevisan [11] and Knight [13]). Jointly considered with (2.7)--(2.8), the estimate yields (4:1) lim inf t!1 (log log t) 1=2 t 1=2 sup 0ut jW (u)jsp 2; a.s. Compared with the classical Chung =-=[6]-=- LIL for W (which says that the liminf term in (4.1) equals = p 8 ), it is noted that (4.1) gives the correct rate function (but not the constant) for the lower behavior of Brownian motion. 4.2. L p -... |

25 |
Sample functions of stochastic processes with stationary, independent increments
- Fristedt
- 1974
(Show Context)
Citation Context ...: Now we just note that t 7! (s + t) \Gamma (s) is the inverse local time for r 7! W (r + (s)) \Gamma W ((s)). The fact that W has iid increments implies that Y does as well. Theorem 11.6 of Fristedt =-=[9]-=- now implies that for some absolute constant c 3 2 (0; 1), (2:6) lim inf t!1 (log log t) 3 t 3 Y (t) = c 3 ; a.s. On the other hand, t 7! (t) is a strictly stable subordinator with index (1=2). For a ... |

20 |
The small ball problem for the Brownian sheet
- Talagrand
- 1994
(Show Context)
Citation Context ...y Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], [7], [15]--[18], [21]--[23], [27]--=-=[31]-=-. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and define (1:2) X(t), Z t 0 W (s) ds; ts0: ... |

18 | Comparison results for the lower tail of Gaussian seminorms - Li - 1992 |

16 |
The integral of symmetric unimodular functions over a symmetric set and some probability inequalities
- Anderson
- 1955
(Show Context)
Citation Context ...1; sup 0sm ? ? ? e X(s) +X(n) + sW (n) \Gamma x ? ? ?! 1 j ; (2:2) where f e X(t); ts0g stands for an independent copy of fX(t); ts0g. By the shifted ball property for Gaussian measures (cf. Anderson =-=[3]-=-), (2:3) P i sup 0st jX(s) \Gamma xj !sjsP i X (t) !sj ; for all t ? 0,s? 0 and x 2 R. Conditioning on fW (t); 0stsng on the right--hand--side of (2.2) yields a n+msP i X (n) ! 1 j P i X (m) ! 1 jsa n... |

15 |
A winding problem for a resonant driven by a white noise
- McKean
- 1963
(Show Context)
Citation Context ...for this example (cf. Remark 2.1 below). (b) Integrated Brownian motion is an interesting process on its own. For example, it appears naturally in classical literature in mechanics and biology ([10], =-=[24]-=-, [26]; cf. Lachal [19] for a detailed account), as well as in the more recent works in quantum probability ([1], [2], [14]). (c) Our analysis of X uses a genuinely new approach to the small ball prob... |

11 |
Some exact equivalents for the Brownian motion in H61der norm, Prob
- Baldi, Roynette
- 1992
(Show Context)
Citation Context ...ence Foundation and the National Security Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works =-=[4]-=-, [7], [15]--[18], [21]--[23], [27]--[31]. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and... |

11 |
Lira inf results for Gaussian samples and Chung's functional
- Kuelbs, Li, et al.
- 1992
(Show Context)
Citation Context ...and the National Security Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], [7], [15]--=-=[18]-=-, [21]--[23], [27]--[31]. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and define (1:2) X(t... |

10 | Some small ball probabilities for Gaussian processes under nonuniform norms - Stolz - 1996 |

9 | Small ball probabilities for Gaussian processes with stationary increments under Hölder norms - Kuelbs, Li, et al. - 1995 |

8 |
Small ball estimates for Gaussian processes under Sobolev type norms
- Li, Shao
- 1999
(Show Context)
Citation Context ...onal Security Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], [7], [15]--[18], [21]--=-=[23]-=-, [27]--[31]. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and define (1:2) X(t), Z t 0 W (... |

7 |
Fubini’s theorem for double Wiener integrals and the variance of the Brownian path
- Donati-Martin, Yor
- 1991
(Show Context)
Citation Context ...herefore possible to prove (3.1) using eigenvalue expansions for Gaussian quadratic functionals. However, by making use of a stochastic Fubini argument which we have learnt from DonatiMartin and Yor (=-=[8]-=-; see also Yor [33, Chap. II]), we can obtain (3.1) without any technical computations. All that we need is the introduction of an auxiliary independent Brownian motion. Indeed, we have E exp i \Gamma... |

6 |
A property of Brownian motion paths
- Trotter
- 1958
(Show Context)
Citation Context ...ion. 5 2.3. Finiteness of c 2 Recalling Eq. (2.4), we propose to verify that c 2 ! 1. To this end, define the local time of W by, L(t; x), d dx Z t 0 I fW (s)!xg ds: According to a theorem of Trotter =-=[32]-=-, a version of (t; x) 7! L(t; x) can be chosen so as to be almost surely continuous. We will work only with this version. Write (s) = inff t ? 0 : L(t; 0) ? s g; s ? 0: This is the right--continuous i... |

4 |
Lim inf results for the Wiener process and its increments under the L2-norm. Probab. Theory Related Fields 92
- Li
- 1992
(Show Context)
Citation Context ...e National Security Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], [7], [15]--[18], =-=[21]-=---[23], [27]--[31]. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and define (1:2) X(t), Z t... |

4 |
Small ball probabilities of Gaussian fields
- Shao, Wang
- 1995
(Show Context)
Citation Context ...ecurity Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], [7], [15]--[18], [21]--[23], =-=[27]-=---[31]. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and define (1:2) X(t), Z t 0 W (s) ds;... |

4 | Small ball probabilities for a Wiener process under weighted sup-norms with an application to the supremum of Bessel local times - Shi - 1996 |

3 |
Small ball problems for Brownian motion and for the Brownian sheet
- Kuelbs, Li
- 1993
(Show Context)
Citation Context ...ation and the National Security Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], [7], =-=[15]-=---[18], [21]--[23], [27]--[31]. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and define (1:... |

2 |
Estimates Uniform in Time for the Transition Probability of Diffusions with Small Drift and for Stochastically Perturbed Newton Equations
- Albeverio, Hilbert, et al.
- 1999
(Show Context)
Citation Context ...ple, it appears naturally in classical literature in mechanics and biology ([10], [24], [26]; cf. Lachal [19] for a detailed account), as well as in the more recent works in quantum probability ([1], =-=[2]-=-, [14]). (c) Our analysis of X uses a genuinely new approach to the small ball problem. We first prove a Chung's law of the iterated logarithm (LIL) for X and then use it to establish the rate of expl... |

2 |
Probability theory: its role and its impact
- Kac
- 1962
(Show Context)
Citation Context ...oblem for this example (cf. Remark 2.1 below). (b) Integrated Brownian motion is an interesting process on its own. For example, it appears naturally in classical literature in mechanics and biology (=-=[10]-=-, [24], [26]; cf. Lachal [19] for a detailed account), as well as in the more recent works in quantum probability ([1], [2], [14]). (c) Our analysis of X uses a genuinely new approach to the small bal... |

2 |
A note on the long time asymptotics of the Brownian motion with application to the theory of quantum measurement, Potential Anal. 7
- Kolokoltsov
- 1997
(Show Context)
Citation Context ...it appears naturally in classical literature in mechanics and biology ([10], [24], [26]; cf. Lachal [19] for a detailed account), as well as in the more recent works in quantum probability ([1], [2], =-=[14]-=-). (c) Our analysis of X uses a genuinely new approach to the small ball problem. We first prove a Chung's law of the iterated logarithm (LIL) for X and then use it to establish the rate of explosion ... |

1 |
Transience of stochastically perturbed classical Hamiltonian systems and random wave operators. (preprint
- Albeverio, Hilbert, et al.
- 1996
(Show Context)
Citation Context ... example, it appears naturally in classical literature in mechanics and biology ([10], [24], [26]; cf. Lachal [19] for a detailed account), as well as in the more recent works in quantum probability (=-=[1]-=-, [2], [14]). (c) Our analysis of X uses a genuinely new approach to the small ball problem. We first prove a Chung's law of the iterated logarithm (LIL) for X and then use it to establish the rate of... |

1 |
Random fractals and Chung-type functional laws of the iterated logarithm. (preprint
- Deheuvels, Mason
- 1996
(Show Context)
Citation Context ...Foundation and the National Security Agency 1 analysis. Known results for well--behaved processes (especially, Brownian motion and related processes) can be found for example in the recent works [4], =-=[7]-=-, [15]--[18], [21]--[23], [27]--[31]. The aim of this paper is to study the special case of integrated Brownian motion. Let fW (t); ts0g be a one--dimensional Brownian motion starting from 0, and defi... |

1 |
L'evy--classes and self--normalization
- Khoshnevisan
- 1996
(Show Context)
Citation Context ...ion (and 10 other processes). For example, by the Brownian excursion theory, it is easily seen that \Gamma sup 0u(t) jW (u)j \Delta \Gamma1 has an exponential law of parameter t; cf. Lemma 2.4 (i) of =-=[11]-=-. By (a careful application of) the Borel--Cantelli lemma, this leads to: lim inf t!1 log log t t sup 0u(t) jW (u)j = 1; a.s. (For details and more general results, cf. Khoshnevisan [11] and Knight [1... |

1 |
Calculation of characteristic functions of some functionals of a Wiener process and a Brownian bridge (English translation
- Klyachko, Solodyannikov
- 1987
(Show Context)
Citation Context ...dt = 1 2 Z 1 0 W (t)W (1 \Gamma t) dt \Gamma 1 2 Z 1 0 f W (t) f W(1 \Gamma t) dt: (3:3) The law of R 1 0 W (t)W (1 \Gamma t)dt is determined by the following result due to Klyachko and Solodyannikov =-=[12]-=-: E exp i i Z 1 0 W (t)W (1 \Gamma t)dt j = i 1 + i cosh ps+ i cos p j 1=2 ; from which (3.2) immediately follows with the aid of (3.3). tu Proof of Theorem 3.1. Clearly, t 7! X(t) is a centered Gauss... |

1 |
Local variation of diffusion in local time
- Knight
- 1973
(Show Context)
Citation Context ...1]. By (a careful application of) the Borel--Cantelli lemma, this leads to: lim inf t!1 log log t t sup 0u(t) jW (u)j = 1; a.s. (For details and more general results, cf. Khoshnevisan [11] and Knight =-=[13]-=-). Jointly considered with (2.7)--(2.8), the estimate yields (4:1) lim inf t!1 (log log t) 1=2 t 1=2 sup 0ut jW (u)jsp 2; a.s. Compared with the classical Chung [6] LIL for W (which says that the limi... |

1 |
Etude des Trajectoires de la Primitive du Mouvement Brownien. Th`ese de Doctorat de l'Universit'e Claude Bernard Lyon I
- Lachal
- 1992
(Show Context)
Citation Context ...emark 2.1 below). (b) Integrated Brownian motion is an interesting process on its own. For example, it appears naturally in classical literature in mechanics and biology ([10], [24], [26]; cf. Lachal =-=[19]-=- for a detailed account), as well as in the more recent works in quantum probability ([1], [2], [14]). (c) Our analysis of X uses a genuinely new approach to the small ball problem. We first prove a C... |

1 |
First exit time from a bounded interval for integrated Brownian motion. (preprint
- Lachal
- 1996
(Show Context)
Citation Context ...j ! 0 was known, cf. Stolz [29, Chap. 5] (it can also be deduced using the argument in the proof of Theorem 1.1 of Kuelbs and Li [15], for example). It is also possible, using Proposition 8 of Lachal =-=[20], to show that -=-lim sup "!0 " 2=3 log P i X (1) ! " js\Gamma 1 e i 6 j 1=3 : Since (=6) 1=3 =e ! 3=8, this is not as sharp as the estimate c 1s3=8 in Remark 1.4. However, as we have pointed out, the ha... |

1 | Mesures Gaussiennes de Petites Boules et Petites D'eviations. Th`ese de Doctorat de l'Universit'e Paul Sabatier Toulouse III - Stolz - 1995 |