Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices
Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, C...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Li, Deli [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
1998 |
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| Anmerkung: |
© Plenum Publishing Corporation 1998 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of theoretical probability - Kluwer Academic Publishers-Plenum Publishers, 1988, 11(1998), 2 vom: Apr., Seite 443-459 |
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| Übergeordnetes Werk: |
volume:11 ; year:1998 ; number:2 ; month:04 ; pages:443-459 |
| Links: |
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| DOI / URN: |
10.1023/A:1022687923455 |
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| Katalog-ID: |
OLC2053621354 |
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| 520 | |a Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. | ||
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10.1023/A:1022687923455 doi (DE-627)OLC2053621354 (DE-He213)A:1022687923455-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Li, Deli verfasserin aut Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. Tomkins, R. J. aut Enthalten in Journal of theoretical probability Kluwer Academic Publishers-Plenum Publishers, 1988 11(1998), 2 vom: Apr., Seite 443-459 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:11 year:1998 number:2 month:04 pages:443-459 https://doi.org/10.1023/A:1022687923455 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2088 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4325 AR 11 1998 2 04 443-459 |
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10.1023/A:1022687923455 doi (DE-627)OLC2053621354 (DE-He213)A:1022687923455-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Li, Deli verfasserin aut Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. Tomkins, R. J. aut Enthalten in Journal of theoretical probability Kluwer Academic Publishers-Plenum Publishers, 1988 11(1998), 2 vom: Apr., Seite 443-459 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:11 year:1998 number:2 month:04 pages:443-459 https://doi.org/10.1023/A:1022687923455 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2088 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4325 AR 11 1998 2 04 443-459 |
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10.1023/A:1022687923455 doi (DE-627)OLC2053621354 (DE-He213)A:1022687923455-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Li, Deli verfasserin aut Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. Tomkins, R. J. aut Enthalten in Journal of theoretical probability Kluwer Academic Publishers-Plenum Publishers, 1988 11(1998), 2 vom: Apr., Seite 443-459 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:11 year:1998 number:2 month:04 pages:443-459 https://doi.org/10.1023/A:1022687923455 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2088 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4325 AR 11 1998 2 04 443-459 |
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10.1023/A:1022687923455 doi (DE-627)OLC2053621354 (DE-He213)A:1022687923455-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Li, Deli verfasserin aut Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. Tomkins, R. J. aut Enthalten in Journal of theoretical probability Kluwer Academic Publishers-Plenum Publishers, 1988 11(1998), 2 vom: Apr., Seite 443-459 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:11 year:1998 number:2 month:04 pages:443-459 https://doi.org/10.1023/A:1022687923455 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2088 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4325 AR 11 1998 2 04 443-459 |
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10.1023/A:1022687923455 doi (DE-627)OLC2053621354 (DE-He213)A:1022687923455-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Li, Deli verfasserin aut Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices 1998 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Plenum Publishing Corporation 1998 Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. Tomkins, R. J. aut Enthalten in Journal of theoretical probability Kluwer Academic Publishers-Plenum Publishers, 1988 11(1998), 2 vom: Apr., Seite 443-459 (DE-627)129930903 (DE-600)357043-5 (DE-576)018191886 0894-9840 nnns volume:11 year:1998 number:2 month:04 pages:443-459 https://doi.org/10.1023/A:1022687923455 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_24 GBV_ILN_40 GBV_ILN_70 GBV_ILN_105 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2006 GBV_ILN_2088 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4315 GBV_ILN_4325 AR 11 1998 2 04 443-459 |
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Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). 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compact laws of the iterated logarithm for b-valued random variables with two-dimensional indices |
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Compact Laws of the Iterated Logarithm for B-Valued Random Variables with Two-Dimensional Indices |
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Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. © Plenum Publishing Corporation 1998 |
| abstractGer |
Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. © Plenum Publishing Corporation 1998 |
| abstract_unstemmed |
Abstract Let $$({\text{B,||}} \cdot {\text{||}})$$ be a real separable Banach space and {X, Xn, m; (n, m) ∈ N2} B-valued i.i.d. random variables. Set $$S(n,m) = \sum\nolimits_{i = 1}^n {\sum\nolimits_{j = 1}^m {X_{i,j} ,(n,m) \in N} } ^2$$. In this paper, the compact law of the iterated logarithm, CLIL(D), for B-valued random variables with two-dimensional indices ranging over a subset D of N2 is studied. There is a gap between the moment conditions for CLIL(N1) and those for CLIL(N2). The main result of this paper fills this gap by presenting necessary and sufficient conditions for the sequence $${{\{ S(n,m)} \mathord{\left/ {\vphantom {{\{ S(n,m)} {\sqrt {2nm\log \log (nm);} (n,m) \in }}} \right. \kern-\nulldelimiterspace} {\sqrt {2nm\log \log (nm);} (n,m) \in }}N^r ({\alpha , }\varphi {)\} }$$ to be almost surely conditionally compact in B, where, for α ≥ 0, 1 ≤ r ≤ 2, Nr(α, φ) = {(n, m) ∈ N2; nα ≤ m ≤ nα exp{(log n)r−1 φ(n)}} and φ(·) is any positive, continuous, nondecreasing function such that φ(t)/(log log t)γ is eventually decreasing as t → ∞, for some γ > 0. © Plenum Publishing Corporation 1998 |
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