## Exponential integrability and transportation cost related to logarithmic Sobolev inequalities (1999)

Venue: | J. FUNCT. ANAL |

Citations: | 95 - 5 self |

### BibTeX

@ARTICLE{Bobkov99exponentialintegrability,

author = {S. G. Bobkov and F. Gotze},

title = {Exponential integrability and transportation cost related to logarithmic Sobolev inequalities},

journal = {J. FUNCT. ANAL},

year = {1999},

volume = {163},

pages = {1--28}

}

### Years of Citing Articles

### OpenURL

### Abstract

We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities.

### Citations

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Citation Context ...mple. Let �� be an arbitrary product probability measure on the cube\Omega = [\Gamma1; 1] n . Recently, M.Ledoux [L3] established, in the spirit of some of Talagrand's concentration inequalies ([T=-=1], [T3]-=-), for such a measure a logarithmic Sobolev inequality (1.1) for the class A of all convex smooth functions on [\Gamma1; 1] n and also found the optimal constant c = 4. Together with (2.5) this gives:... |

256 |
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Citation Context ...s to be optimal for the choice \Phi(x) = jxj. For \Phi(x) = e x , this inequality yields Z R n e f dfl nsZ R n exp ` �� 2 8 jrf j 2 ' dfl n : (2.6) On the other hand, by Gross' logarithmic inequal=-=ity [G1]-=-, (1.2) holds for fl n with c = 1. Therefore, according to (2.5), the constant in the exponent can somewhat be improved: Corollary 2.2 For any integrable smooth function f on R n with R f dfl n = 0, Z... |

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Citation Context ...e a characterization of those probablity measures which satisfy these inequalities. 1 Introduction Logarithmic Sobolev inequalities are an essential tool in the study of many problems (cf. e.g. [G2], =-=[D-SC]-=-, [L3]). The main purpose of the present notes is to refine some known connections between logarithmic Sobolev inequalities, exponential integrability of 'smooth' functions and the concentration of me... |

112 |
Optimal numberings and isoperimetric problems on graphs
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Citation Context ...ties, (3.3) is apparently the sharpest one. The particular two-point case\Omega = f0; 1g n with uniform measure �� is the exception: the left-hand side of (3.3) is known and given by Harper's theo=-=rem [H]. An-=-other important example is the Gaussian measure �� = fl n on\Omega = R n with the usual Euclidean metric. In this case, the inequality (3.1), with c = 1, has been recently established by M.Talagra... |

88 | On Talagrand’s deviation inequalities for product measures
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Citation Context ...acterization of those probablity measures which satisfy these inequalities. 1 Introduction Logarithmic Sobolev inequalities are an essential tool in the study of many problems (cf. e.g. [G2], [D-SC], =-=[L3]-=-). The main purpose of the present notes is to refine some known connections between logarithmic Sobolev inequalities, exponential integrability of 'smooth' functions and the concentration of measure.... |

74 |
Ultracontractivity and the heat kernels for Schrodinger operators and Dirichlet
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Citation Context ... 2c : (2.4) The inequality (2.3) has been shown to follow from (1.1) by M.Ledoux [L1], [L2], and (2.4) was deduced from (1.1) by S.Aida, T.Masuda and I.Shigekawa [A-M-S], see also [A-S] (according to =-=[D-S]-=-, the original idea goes back to I.Herbst). The above deduction of (2.4) from (2.2) in essense repeates an argument of [A-M-S]. As for the proof of (2.1), we develop an argument of M.Ledoux which was ... |

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Citation Context ...' functions similar to the ones due to S.Aida, T.Masuda and I.Shigekawa [A-M-S] and M.Ledoux [L1], [L2]. In the Gaussian case, (1.3) with ff = c = 1 improves an exponential inequality due to G.Pisier =-=[P1]-=-. In the case of the discrete cube, it affirmatively answers his question (p.182) on the validity of the discrete analogue of the Gaussian variant of (1.3). These applications will be discussed in mor... |

72 |
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Citation Context ...1 \Gamma ��(A h )sexp 0 @ \Gamma 1 2c / h \Gamma s 2c log 1 ��(A) ! 2 1 A ; (1.5) where A h denotes h-neighbourhood of a set A ae \Omega\Gamma This approach has been studied as well by M.Talag=-=rand in [T4] pro-=-ving, in particular, that the canonical Gaussian measure �� = fl n on the Euclidean space satisfies an inequality (1.4) for the W 2 -metric: W 2 (fl n ; )ss 2 Z log d dfl n d: (1.6) As a consequen... |

68 |
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Citation Context ...en give a characterization of those probablity measures which satisfy these inequalities. 1 Introduction Logarithmic Sobolev inequalities are an essential tool in the study of many problems (cf. e.g. =-=[G2]-=-, [D-SC], [L3]). The main purpose of the present notes is to refine some known connections between logarithmic Sobolev inequalities, exponential integrability of 'smooth' functions and the concentrati... |

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Citation Context ...(x) = x 2 log(1 + x 2 ). Thus, specializing to the real line\Omega = R, we reduce LSI c to a Hardy-type inequality. Finally, we use (in section 5) a result of M.Artola, G.Talenti and G.Tomaselli (cf. =-=[Mu]) to derive -=-necessary and sufficient conditions for the measure �� on R to satisfy LSI c for some finite c. More precisely, let F (x) = ��((\Gamma1; x]), x 2 R, denote the distribution function of ��,... |

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Citation Context .... The question of a functional representation for metrics of Kantorovich-Rubinshteintype (raised by R.M.Dudley) was open for some time until V.L.Levin and S.T.Rachev proved in particular that ([Lev], =-=[Ra]) W 2 2 -=-(; ��) = sup Z g d \Gamma Z f d��; LOGARITHMIC SOBOLEV INEQUALITIES 4 where the supremum is taken over all pairs of bounded continous functions (g; f) such that g(y) \Gamma f(x)sd(x; y) 2 , fo... |

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Citation Context ...process with variances oe 2 t . In order to prove (3.15), we will follow an argument of B.Maurey [Mau]. We use the Brunn-Minkowki inequality in the functional form of A.Pr'ekopa and L.Leindler ([Pr], =-=[Lei]-=-, cf. also [P2, p.3] for a simple proof due to K.Ball): given (fixed)s2 (0; 1), for any measurable functions u, v and w on R n , `Z e \Gammau dx 's`Z e \Gammav dx ' 1\Gamma Z e \Gammaw dx (3.16) holds... |

36 |
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Citation Context ...lowing up lemma of R.Ahlswede, P.G'acs and J.Korner [A-G-K], she established concentration inequalities for discrete product measures (and, furthermore, for distributions of certain Markov processes, =-=[Ma2]), 1 \Ga-=-mma ��(A h )sexp 0 @ \Gamma 1 2c / h \Gamma s 2c log 1 ��(A) ! 2 1 A ; (1.5) where A h denotes h-neighbourhood of a set A ae \Omega\Gamma This approach has been studied as well by M.Talagrand ... |

29 | Bounds on conditional probabilities with applications in multi-user communication. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 34:157–177
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Citation Context ...ormational divergence D( k ��) = R log d d�� d: Such transportation inequalities have been introduced for ff = 1 by K.Marton [Ma1]. Sharpening the blowing up lemma of R.Ahlswede, P.G'acs and J=-=.Korner [A-G-K], sh-=-e established concentration inequalities for discrete product measures (and, furthermore, for distributions of certain Markov processes, [Ma2]), 1 \Gamma ��(A h )sexp 0 @ \Gamma 1 2c / h \Gamma s ... |

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Citation Context ...(and with c = 1) by a relation between the distributions of an arbitrary function f and (Sf)(x) = inf ae f(y) + 1 2 d(x; y) 2 : y 2\Omega oe ; that is, Z e Sf d��se R f d�� : (1.7) Following B=-=.Mauray [Mau]-=- who introduced these inequalities as a functional approach to some M.Talagrand's isoperimetric inequalities ([T1], [T2]), they are now referred to as inf-convolution inequalities. Using this approach... |

29 |
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Citation Context ...y) 2 : y 2\Omega oe ; that is, Z e Sf d��se R f d�� : (1.7) Following B.Mauray [Mau] who introduced these inequalities as a functional approach to some M.Talagrand's isoperimetric inequalities=-= ([T1], [T2]), t-=-hey are now referred to as inf-convolution inequalities. Using this approach, we shall give a simple alternative proof of Talagrand's transportation inequality (1.6) via (1.7) with �� = fl n . In ... |

26 |
A simple proof of the blowing-up lemma
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Citation Context ...inimal transportation cost needed to transportsinto �� to the so-called informational divergence D( k ��) = R log d d�� d: Such transportation inequalities have been introduced for ff = 1 =-=by K.Marton [Ma1]-=-. Sharpening the blowing up lemma of R.Ahlswede, P.G'acs and J.Korner [A-G-K], she established concentration inequalities for discrete product measures (and, furthermore, for distributions of certain ... |

21 |
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Citation Context ... f # A with f d+=0, | e f d+ | ec1( f )2 d+. (1.3) This estimate implies inequalities for exponential moments of ``Lipschitz'' functions similar to those proved by S. Aida, T. Masuda and I. Shigekawa =-=[A-M-S]-=- and M. Ledoux [L1], [L2]. In the Gaussian case, (1.3) improves an exponential inequality due to G. Pisier [P1]. In the case of4 BOBKOV AND GOTZE the discrete cube, it affirmatively answers his quest... |

17 |
estimates derived from Poincaré and logarithmic Sobolev inequalities
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Citation Context ... 2 d�� ' ; 0 ! t ! 1 2c : (2.4) The inequality (2.3) has been shown to follow from (1.1) by M.Ledoux [L1], [L2], and (2.4) was deduced from (1.1) by S.Aida, T.Masuda and I.Shigekawa [A-M-S], see a=-=lso [A-S]-=- (according to [D-S], the original idea goes back to I.Herbst). The above deduction of (2.4) from (2.2) in essense repeates an argument of [A-M-S]. As for the proof of (2.1), we develop an argument of... |

17 |
Isoperimetry and Gaussian Analysis. École d’été de Probabilités de Saint-Flour, LNM 1648
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Citation Context ... d�� ' c=(2ff\Gammac) : (1.3) This estimate implies some inequalities for exponential moments of 'Lipschitz' functions similar to the ones due to S.Aida, T.Masuda and I.Shigekawa [A-M-S] and M.Led=-=oux [L1]-=-, [L2]. In the Gaussian case, (1.3) with ff = c = 1 improves an exponential inequality due to G.Pisier [P1]. In the case of the discrete cube, it affirmatively answers his question (p.182) on the vali... |

15 |
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Citation Context ...unction). Proof of Proposition 4.1. One may assume that f 2 LN , kfkN = 1, and that R fd�� = 0. The second inequality in (4.3) is essentially a version of an inequality due to O.S.Rothaus who show=-=ed ([Ro], Lemma 10) that-=- L(f)sEnt (f 2 ) + 2 Z f 2 d�� (4.7) whenever R fd�� = 0. Indeed, introducing the function U(x) = 2x \Gamma x log x, xs0, and noting that Z f 2 log f 2 d��sZ f 2 log(1 + f 2 )d�� = Z N... |

5 |
On Gross' and Talagrand's inequalities on the discrete cube
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- 1995
(Show Context)
Citation Context ...; 1g n . In this case, (1.1) holds true for convex functions with a better constant c = 2. This can easily be shown using the discrete Gross' logarithmic inequality [G1] (such an argument was used in =-=[B]-=-). Thus the constant 4 in the exponent in (2.8) can be replaced by 2. Now we are able to state a discrete version of Pisier's inequality (2.6). Take for \Gamma(f ) = D(f) the length of discrete gradie... |

3 |
Remarks on logarithmic Sobolev constants, exponential integrability and bounds on the diameter
- Ledoux
- 1995
(Show Context)
Citation Context ... c=(2ff\Gammac) : (1.3) This estimate implies some inequalities for exponential moments of 'Lipschitz' functions similar to the ones due to S.Aida, T.Masuda and I.Shigekawa [A-M-S] and M.Ledoux [L1], =-=[L2]-=-. In the Gaussian case, (1.3) with ff = c = 1 improves an exponential inequality due to G.Pisier [P1]. In the case of the discrete cube, it affirmatively answers his question (p.182) on the validity o... |

3 |
The problem of mass transfer in a topological space, and probability measures having given marginal measures on the product of two spaces
- Levin
- 1984
(Show Context)
Citation Context ...r W ff . The question of a functional representation for metrics of Kantorovich-Rubinshteintype (raised by R.M.Dudley) was open for some time until V.L.Levin and S.T.Rachev proved in particular that (=-=[Lev], [Ra]) -=-W 2 2 (; ��) = sup Z g d \Gamma Z f d��; LOGARITHMIC SOBOLEV INEQUALITIES 4 where the supremum is taken over all pairs of bounded continous functions (g; f) such that g(y) \Gamma f(x)sd(x; y) ... |

3 |
private communication
- Talagrand
- 2003
(Show Context)
Citation Context ...s necessary to prove this claim. Thus, it is natural to ask whether or not, even for one dimension, the above implication holds for ff = 1=2. This turns out be true (an observation due to M.Talagrand =-=[T5]). C-=-onsider another important example. Let �� be an arbitrary product probability measure on the cube\Omega = [\Gamma1; 1] n . Recently, M.Ledoux [L3] established, in the spirit of some of Talagrand's... |

2 |
An isoperimetric theorem on the cube and the Khinchine-Kahane inequalities
- Talagrand
- 1988
(Show Context)
Citation Context ... d(x; y) 2 : y 2\Omega oe ; that is, Z e Sf d��se R f d�� : (1.7) Following B.Mauray [Mau] who introduced these inequalities as a functional approach to some M.Talagrand's isoperimetric inequa=-=lities ([T1], [T-=-2]), they are now referred to as inf-convolution inequalities. Using this approach, we shall give a simple alternative proof of Talagrand's transportation inequality (1.6) via (1.7) with �� = fl n... |

1 |
Logarithmic Sobolev inequalties and exponential integrability
- Aida, Masuda, et al.
- 1994
(Show Context)
Citation Context ...s`Z e ff \Gamma(f ) 2 d�� ' c=(2ff\Gammac) : (1.3) This estimate implies some inequalities for exponential moments of 'Lipschitz' functions similar to the ones due to S.Aida, T.Masuda and I.Shigek=-=awa [A-M-S]-=- and M.Ledoux [L1], [L2]. In the Gaussian case, (1.3) with ff = c = 1 improves an exponential inequality due to G.Pisier [P1]. In the case of the discrete cube, it affirmatively answers his question (... |

1 | Real analisis and probability - Dudley - 1989 |

1 |
On logarithmic concave measures and functions
- Pr'ecopa
- 1973
(Show Context)
Citation Context ...ssian process with variances oe 2 t . In order to prove (3.15), we will follow an argument of B.Maurey [Mau]. We use the Brunn-Minkowki inequality in the functional form of A.Pr'ekopa and L.Leindler (=-=[Pr]-=-, [Lei], cf. also [P2, p.3] for a simple proof due to K.Ball): given (fixed)s2 (0; 1), for any measurable functions u, v and w on R n , `Z e \Gammau dx 's`Z e \Gammav dx ' 1\Gamma Z e \Gammaw dx (3.16... |

1 |
A geometric approach to a maximum likelyhood estimation for infinite-dimensional Gaussian location
- Tsirel'son
- 1985
(Show Context)
Citation Context ...alagrand 's transportation inequality (3.12). It might be worthwile to note that, for convex functions f on R n , (3.15) can be rewritten in an infinite dimensional setting as Tsirel'son's inequality =-=[Ts]-=- E expfsup t (x t \Gamma oe 2 t =2)gsexpfE sup t x t g; where x t denotes an arbitrary bounded Gaussian process with variances oe 2 t . In order to prove (3.15), we will follow an argument of B.Maurey... |

1 |
Logarithmic Sobolev inequalities for finite Marlov chains
- Diaconis, Saloff-Coste
- 1996
(Show Context)
Citation Context ...nential integrability; concentration of measure; transportation inequalities. 1. INTRODUCTION Logarithmic Sobolev inequalities are an essential tool in the study of various problems (cf., e.g., [G2], =-=[D-SC]-=-, [L3]). The main purpose of this paper is to refine some known connections between logarithmic Sobolev inequalities, exponential integrability of ``smooth'' functions and the concentration of measure... |

1 |
Optimal numbering and isopermimetric problems on graphs
- Harper
- 1966
(Show Context)
Citation Context ... apparently the sharpest one. The particular two-point case 0=[0, 1] n with uniform measure + is the exception: The optimal bound on the left-hand side of (3.3) is known and given by Harper's theorem =-=[H]-=-. Another important example is the Gaussian measure +=# n on 0=R n with the usual Euclidean metric. In this case, the inequality (3.1), with c=1, has recently been established by M. Talagrand [T4]. Hi... |