zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Local time and related sample paths of filtered white noises. (English) Zbl 1144.60029
The author studies the properties of the local times of the Filtered White Noise (FWN), which is a Gaussian process with representation $$ X(t)=\int_{\mathbb{R}} \frac{a(t,\lambda)(e^{it\lambda}-1)}{\vert \lambda\vert ^{1/2+H}}dW(\lambda),\quad t\in [0,1], $$ where $0<H<1$. The main result of the paper states that FWN has (a.s) jointly continuous local time $L(t,x)$ which is Hölder continuous, with certain Hölder exponents. The approach used in the proof is based on the Berman concept of local nondeterminism, see [{\it S. M. Berman}, Ann. Math. Stat. 41, 1260--1272 (1970; Zbl 0204.50501]. Further the author proves the Chung’s law of iterated logarithm for FWN, and uses this result to find the lower bound for the moduli of continuity of the local time. See also the related papers [{\it B. Boufoussi, M. Dozzi, R. Guerbaz}, Bernoulli 13, No. 3, 849--867 (2007; Zbl 1138.60032) and Stochastics 78, No. 1, 33--49 (2006; Zbl 1124.60061)].

MSC:
60G15Gaussian processes
60G17Sample path properties
WorldCat.org
Full Text: DOI Numdam EuDML
References:
[1] R. Adler. , Geometry of random fields , Wily , 1980 MR 611857 | Zbl 0478.60059 · Zbl 0478.60059
[2] A. Benassi , Serge Cohen and Jacques Istas , Local self similarity and Hausdorff dimension , C.R.A.S. , Série I, tome 336 : 267 - 272 , 2003 MR 1968271 | Zbl 1023.60043 · Zbl 1023.60043 · doi:10.1016/S1631-073X(03)00015-3
[3] A. Benassi , Serge Cohen , Jacques Istas and S. Jaffard , Identification of filtered white noises , Stochastic Processes and their Applications , 75 : 31 - 49 , 1998 MR 1629014 | Zbl 0932.60037 · Zbl 0932.60037 · doi:10.1016/S0304-4149(97)00123-3
[4] S. M. Berman , Gaussian processes with stationary increments: Local times and sample function properties , Ann. Math. Statist. , 41 : 1260 - 1272 , 1970 MR 272035 | Zbl 0204.50501 · Zbl 0204.50501 · doi:10.1214/aoms/1177696901
[5] S. M. Berman , Gaussian sample functions: uniform dimension and Hölder conditions nowhere , Nagoya Math. J. , 46 : 63 - 86 , 1972 Article | MR 307320 | Zbl 0246.60038 · Zbl 0246.60038 · http://minidml.mathdoc.fr/cgi-bin/location?id=00079032
[6] S. M. Berman , Local nondeterminism and local times of Gaussian processes , Indiana Univ. Math. J. , 23 : 69 - 94 , 1973 MR 317397 | Zbl 0264.60024 · Zbl 0264.60024 · doi:10.1512/iumj.1973.23.23006
[7] B. Boufoussi , M. Dozzi and R. Guerbaz , On the local time of the multifractional Brownian motion , Stochastics and stochastic repports , 78 , 33-49 MR 2219711 | Zbl 1124.60061 · Zbl 1124.60061 · doi:10.1080/17442500600578073
[8] W. Ehm , Sample function properties of multi-parameter stable processes , Z. Wahrsch. verw. Gebiete , 56 , 195-228 MR 618272 | Zbl 0471.60046 · Zbl 0471.60046 · doi:10.1007/BF00535741
[9] D. Geman and J. Horowitz , Occupation densities , Ann. of Probab. , 8 : 1 - 67 , 1980 MR 556414 | Zbl 0499.60081 · Zbl 0499.60081 · doi:10.1214/aop/1176994824
[10] W. Li and Q. M. Shao , Gaussian Processes : Inequalities, Small Ball Probabilities and Applications , Stochastic Processes: Theory and methods. Handbook of Statistics, 19 , Edited by C.R. Rao and D. Shanbhag Elsevier, New York , 2001 MR 1861734 | Zbl 0987.60053 · Zbl 0987.60053
[11] D. Monrad and H. Rootzén , Small values of Gaussian processes and functional laws of the iterated logarithm , Probab. Th. Rel. Fields , 101 : 173 - 192 , 1995 MR 1318191 | Zbl 0821.60043 · Zbl 0821.60043 · doi:10.1007/BF01375823
[12] L.D. Pitt , Local times for Gaussian vector fields , Indiana Univ. Math. J. , 27 : 309 - 330 , 1978 MR 471055 | Zbl 0382.60055 · Zbl 0382.60055 · doi:10.1512/iumj.1978.27.27024
[13] M. Priestley , Evolutionary spectra and non stationary processes , J. Roy. Statist. Soc. , B 27 : 204 - 237 , 1965 MR 199886 | Zbl 0144.41001 · Zbl 0144.41001
[14] Y. Xiao , Strong local nondeterminism and the sample path properties of Gaussian random fields , Preprint 2005