An improvement of Hoffmann-Jorgensen's inequality
(2000) In Annals of Probability 28(2). p.851-862- Abstract
- Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp... (More)
- Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/record/1766847
- author
- Klass, Michael J. and Nowicki, Krzysztof ^{LU}
- organization
- publishing date
- 2000
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- expo- nential inequalities, Tail probability inequalities, Hoffmann-Jorgensen's inequality, Banach space valued random variables
- in
- Annals of Probability
- volume
- 28
- issue
- 2
- pages
- 851 - 862
- publisher
- Institute of Mathematical Statistics
- ISSN
- 0091-1798
- language
- English
- LU publication?
- yes
- id
- 743fac1c-6501-412a-a36e-3addac2f0c0b (old id 1766847)
- date added to LUP
- 2011-01-27 13:05:09
- date last changed
- 2016-04-16 06:42:21
@article{743fac1c-6501-412a-a36e-3addac2f0c0b, abstract = {Let B be a Banach space and F any family of bounded linear functionals on B of norm at most one. For x ∈ B set || x || = supΛ∈F Λ (x) (||· || is at least a seminorm on B). We give probability estimates for the tail probability of S* n = max1≤ k≤ n ||Σk j=1 Xj || where {Xi}n i=1 are independent symmetric Banach space valued random elements. Our method is based on approximating the probability that S* n exceeds a threshold defined in terms of Σk j=1 Y(j), where Y(r) denotes the rth largest term of {|| Xi ||}n i=1. Using these tail estimates, essentially all the known results concerning the order of magnitude or finiteness of quantities such as EΦ(|| Sn ||) and EΦ(S* n) follow (for any fixed 1 ≤ n ≤ ∞). Included in this paper are uniform Lp bounds of S* n which are within a factor of 4 for all p ≥ 1 and within a factor of 2 in the limit as p → ∞.}, author = {Klass, Michael J. and Nowicki, Krzysztof}, issn = {0091-1798}, keyword = {expo- nential inequalities,Tail probability inequalities,Hoffmann-Jorgensen's inequality,Banach space valued random variables}, language = {eng}, number = {2}, pages = {851--862}, publisher = {Institute of Mathematical Statistics}, series = {Annals of Probability}, title = {An improvement of Hoffmann-Jorgensen's inequality}, volume = {28}, year = {2000}, }