# Law of large numbers and central limit theorem under nonlinear expectations

Ontology type: schema:ScholarlyArticle      Open Access: True

### Article Info

DATE

2019-04-16

AUTHORS ABSTRACT

The main achievement of this paper is the finding and proof of Central Limit Theorem (CLT, see Theorem 12) under the framework of sublinear expectation. Roughly speaking under some reasonable assumption, the random sequence {1/n(X1+⋯+Xn)}i=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{1/\sqrt {n}(X_{1}+\cdots +X_{n})\}_{i=1}^{\infty }$\end{document} converges in law to a nonlinear normal distribution, called G-normal distribution, where {Xi}i=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{X_{i}\}_{i=1}^{\infty }$\end{document} is an i.i.d. sequence under the sublinear expectation. It’s known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties. Under such situation, this new CLT plays a similar role as the one of classical CLT. The classical CLT can be also directly obtained from this new CLT, since a linear expectation is a special case of sublinear expectations. A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT. This paper is originally exhibited in arXiv.(math.PR/0702358v1). More... »

PAGES

4

### Identifiers

URI

http://scigraph.springernature.com/pub.10.1186/s41546-019-0038-2

DOI

http://dx.doi.org/10.1186/s41546-019-0038-2

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1113474463

Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
Incoming Citations Browse incoming citations for this publication using opencitations.net

JSON-LD is the canonical representation for SciGraph data.

TIP: You can open this SciGraph record using an external JSON-LD service:

[
{
"@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json",
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/18",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Law and Legal Studies",
"type": "DefinedTerm"
},
{
"id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/1801",
"inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/",
"name": "Law",
"type": "DefinedTerm"
}
],
"author": [
{
"affiliation": {
"alternateName": "Institute of Mathematics, Shandong University, 250100, Jinan, Shandong Province, China",
"id": "http://www.grid.ac/institutes/grid.27255.37",
"name": [
"Institute of Mathematics, Shandong University, 250100, Jinan, Shandong Province, China"
],
"type": "Organization"
},
"familyName": "Peng",
"givenName": "Shige",
"id": "sg:person.012375343637.10",
"sameAs": [
"https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012375343637.10"
],
"type": "Person"
}
],
"citation": [
{
"id": "sg:pub.10.1007/978-3-540-70847-6_25",
"sameAs": [
"https://app.dimensions.ai/details/publication/pub.1032023006",
"https://doi.org/10.1007/978-3-540-70847-6_25"
],
"type": "CreativeWork"
}
],
"datePublished": "2019-04-16",
"datePublishedReg": "2019-04-16",
"description": "The main achievement of this paper is the finding and proof of Central Limit Theorem (CLT, see Theorem 12) under the framework of sublinear expectation. Roughly speaking under some reasonable assumption, the random sequence {1/n(X1+\u22ef+Xn)}i=1\u221e\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\{1/\\sqrt {n}(X_{1}+\\cdots +X_{n})\\}_{i=1}^{\\infty }$\\end{document} converges in law to a nonlinear normal distribution, called G-normal distribution, where {Xi}i=1\u221e\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}$\\{X_{i}\\}_{i=1}^{\\infty }$\\end{document} is an i.i.d. sequence under the sublinear expectation. It\u2019s known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties. Under such situation, this new CLT plays a similar role as the one of classical CLT. The classical CLT can be also directly obtained from this new CLT, since a linear expectation is a special case of sublinear expectations. A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT. This paper is originally exhibited in arXiv.(math.PR/0702358v1).",
"genre": "article",
"id": "sg:pub.10.1186/s41546-019-0038-2",
"inLanguage": "en",
"isAccessibleForFree": true,
"isPartOf": [
{
"id": "sg:journal.1290466",
"issn": [
"2095-9672",
"2367-0126"
],
"name": "Probability, Uncertainty and Quantitative Risk",
"publisher": "American Institute of Mathematical Sciences (AIMS)",
"type": "Periodical"
},
{
"issueNumber": "1",
"type": "PublicationIssue"
},
{
"type": "PublicationVolume",
}
],
"keywords": [
"central limit theorem",
"sublinear expectations",
"classical CLT",
"new CLT",
"limit theorem",
"nonlinear parabolic PDEs",
"normal distribution",
"parabolic PDEs",
"nonlinear expectations",
"regularity estimates",
"probability measure",
"non-negligible uncertainties",
"linear expectation",
"special case",
"theorem",
"random sequence",
"reasonable assumptions",
"CLT",
"PDEs",
"proof",
"converges",
"law",
"large number",
"such situations",
"main achievements",
"arXiv",
"distribution",
"uncertainty",
"framework",
"assumption",
"estimates",
"situation",
"order",
"number",
"sequence",
"cases",
"expectations",
"similar role",
"important role",
"measures",
"achievement",
"role",
"findings",
"paper"
],
"name": "Law of large numbers and central limit theorem under nonlinear expectations",
"pagination": "4",
"productId": [
{
"name": "dimensions_id",
"type": "PropertyValue",
"value": [
"pub.1113474463"
]
},
{
"name": "doi",
"type": "PropertyValue",
"value": [
"10.1186/s41546-019-0038-2"
]
}
],
"sameAs": [
"https://doi.org/10.1186/s41546-019-0038-2",
"https://app.dimensions.ai/details/publication/pub.1113474463"
],
"sdDataset": "articles",
"sdDatePublished": "2022-05-20T07:35",
"sdPublisher": {
"name": "Springer Nature - SN SciGraph project",
"type": "Organization"
},
"sdSource": "s3://com-springernature-scigraph/baseset/20220519/entities/gbq_results/article/article_796.jsonl",
"type": "ScholarlyArticle",
"url": "https://doi.org/10.1186/s41546-019-0038-2"
}
]

HOW TO GET THIS DATA PROGRAMMATICALLY:

JSON-LD is a popular format for linked data which is fully compatible with JSON.

curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1186/s41546-019-0038-2'

N-Triples is a line-based linked data format ideal for batch operations.

curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1186/s41546-019-0038-2'

curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1186/s41546-019-0038-2'

RDF/XML is a standard XML format for linked data.

curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1186/s41546-019-0038-2'

This table displays all metadata directly associated to this object as RDF triples.

106 TRIPLES      22 PREDICATES      70 URIs      61 LITERALS      6 BLANK NODES

Subject Predicate Object
2 anzsrc-for:1801
3 schema:author N9ae92bb8037149d0af67983dda4797b2
4 schema:citation sg:pub.10.1007/978-3-540-70847-6_25
5 schema:datePublished 2019-04-16
6 schema:datePublishedReg 2019-04-16
7 schema:description The main achievement of this paper is the finding and proof of Central Limit Theorem (CLT, see Theorem 12) under the framework of sublinear expectation. Roughly speaking under some reasonable assumption, the random sequence {1/n(X1+⋯+Xn)}i=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{1/\sqrt {n}(X_{1}+\cdots +X_{n})\}_{i=1}^{\infty }$\end{document} converges in law to a nonlinear normal distribution, called G-normal distribution, where {Xi}i=1∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{X_{i}\}_{i=1}^{\infty }$\end{document} is an i.i.d. sequence under the sublinear expectation. It’s known that the framework of sublinear expectation provides a important role in situations that the probability measure itself has non-negligible uncertainties. Under such situation, this new CLT plays a similar role as the one of classical CLT. The classical CLT can be also directly obtained from this new CLT, since a linear expectation is a special case of sublinear expectations. A deep regularity estimate of 2nd order fully nonlinear parabolic PDE is applied to the proof of the CLT. This paper is originally exhibited in arXiv.(math.PR/0702358v1).
8 schema:genre article
9 schema:inLanguage en
10 schema:isAccessibleForFree true
11 schema:isPartOf N2cdbde36659f4e81beee5b90f5b106f1
12 Ndbb2c306e90445249dfdb8c23e1d07dc
13 sg:journal.1290466
14 schema:keywords CLT
15 PDEs
16 achievement
17 arXiv
18 assumption
19 cases
20 central limit theorem
21 classical CLT
22 converges
23 distribution
24 estimates
25 expectations
26 findings
27 framework
28 important role
29 large number
30 law
31 limit theorem
32 linear expectation
33 main achievements
34 measures
35 new CLT
36 non-negligible uncertainties
37 nonlinear expectations
38 nonlinear parabolic PDEs
39 normal distribution
40 number
41 order
42 paper
43 parabolic PDEs
44 probability measure
45 proof
46 random sequence
47 reasonable assumptions
48 regularity estimates
49 role
50 sequence
51 similar role
52 situation
53 special case
54 sublinear expectations
55 such situations
56 theorem
57 uncertainty
58 schema:name Law of large numbers and central limit theorem under nonlinear expectations
59 schema:pagination 4
60 schema:productId N4019b8304b1841308c845a2677f2656d
61 N4fc6eef0866246c3894a1d54e6d99295
62 schema:sameAs https://app.dimensions.ai/details/publication/pub.1113474463
63 https://doi.org/10.1186/s41546-019-0038-2
64 schema:sdDatePublished 2022-05-20T07:35
66 schema:sdPublisher Nf6b43e2881c04b718209da09461362d0
67 schema:url https://doi.org/10.1186/s41546-019-0038-2
69 sgo:sdDataset articles
70 rdf:type schema:ScholarlyArticle
71 N2cdbde36659f4e81beee5b90f5b106f1 schema:issueNumber 1
72 rdf:type schema:PublicationIssue
73 N4019b8304b1841308c845a2677f2656d schema:name dimensions_id
74 schema:value pub.1113474463
75 rdf:type schema:PropertyValue
76 N4fc6eef0866246c3894a1d54e6d99295 schema:name doi
77 schema:value 10.1186/s41546-019-0038-2
78 rdf:type schema:PropertyValue
79 N9ae92bb8037149d0af67983dda4797b2 rdf:first sg:person.012375343637.10
80 rdf:rest rdf:nil
82 rdf:type schema:PublicationVolume
83 Nf6b43e2881c04b718209da09461362d0 schema:name Springer Nature - SN SciGraph project
84 rdf:type schema:Organization
85 anzsrc-for:18 schema:inDefinedTermSet anzsrc-for:
86 schema:name Law and Legal Studies
87 rdf:type schema:DefinedTerm
88 anzsrc-for:1801 schema:inDefinedTermSet anzsrc-for:
89 schema:name Law
90 rdf:type schema:DefinedTerm
91 sg:journal.1290466 schema:issn 2095-9672
92 2367-0126
93 schema:name Probability, Uncertainty and Quantitative Risk
94 schema:publisher American Institute of Mathematical Sciences (AIMS)
95 rdf:type schema:Periodical
96 sg:person.012375343637.10 schema:affiliation grid-institutes:grid.27255.37
97 schema:familyName Peng
98 schema:givenName Shige
99 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012375343637.10
100 rdf:type schema:Person
101 sg:pub.10.1007/978-3-540-70847-6_25 schema:sameAs https://app.dimensions.ai/details/publication/pub.1032023006
102 https://doi.org/10.1007/978-3-540-70847-6_25
103 rdf:type schema:CreativeWork
104 grid-institutes:grid.27255.37 schema:alternateName Institute of Mathematics, Shandong University, 250100, Jinan, Shandong Province, China
105 schema:name Institute of Mathematics, Shandong University, 250100, Jinan, Shandong Province, China
106 rdf:type schema:Organization