It is shown that functions defined on {0, 1, …, r−1}n satisfying certain conditions of bounded differences that guarantee sub-Gaussian tail behavior also satisfy a much stronger “local” sub-Gaussian property. For self-bounding and configuration functions we derive analogous locally subexponential behavior. The key tool is Talagrand’s [Ann. Probab. 22 (1994) 1576–1587] variance inequality for functions defined on the binary hypercube which we extend to functions of uniformly distributed random variables defined on {0, 1, …, r−1}n for r≥2.
References
Aldous, D. (2001). The $\zeta(2)$ limit in the random assignment problem. Random Structures Algorithms 18 381–418. MR1839499 10.1002/rsa.1015 Aldous, D. (2001). The $\zeta(2)$ limit in the random assignment problem. Random Structures Algorithms 18 381–418. MR1839499 10.1002/rsa.1015
Alon, N., Dinur, I., Friedgut, E. and Sudakov, B. (2004). Graph products, Fourier analysis and spectral techniques. Geom. Funct. Anal. 14 913–940. MR2105948 1056.05104 10.1007/s00039-004-0478-3 Alon, N., Dinur, I., Friedgut, E. and Sudakov, B. (2004). Graph products, Fourier analysis and spectral techniques. Geom. Funct. Anal. 14 913–940. MR2105948 1056.05104 10.1007/s00039-004-0478-3
Alon, N., Krivelevich, M. and Vu, V. H. (2002). On the concentration of eigenvalues of random symmetric matrices. Israel J. Math. 131 259–267. MR1942311 1014.15016 10.1007/BF02785860 Alon, N., Krivelevich, M. and Vu, V. H. (2002). On the concentration of eigenvalues of random symmetric matrices. Israel J. Math. 131 259–267. MR1942311 1014.15016 10.1007/BF02785860
Beckner, W. (1975). Inequalities in Fourier analysis. Ann. Math. 102 159–182. MR0385456 0338.42017 10.2307/1970980 Beckner, W. (1975). Inequalities in Fourier analysis. Ann. Math. 102 159–182. MR0385456 0338.42017 10.2307/1970980
Benjamini, I., Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance. Ann. Probab. 31 1970–1978. MR2016607 1087.60070 10.1214/aop/1068646373 euclid.aop/1068646373 Benjamini, I., Kalai, G. and Schramm, O. (2003). First passage percolation has sublinear distance variance. Ann. Probab. 31 1970–1978. MR2016607 1087.60070 10.1214/aop/1068646373 euclid.aop/1068646373
Bonami, A. (1970). Étude des coefficients de Fourier des fonctions de $L\sp{p}(G)$. Ann. Inst. Fourier (Grenoble) 20 335–402. MR0283496 10.5802/aif.357 Bonami, A. (1970). Étude des coefficients de Fourier des fonctions de $L\sp{p}(G)$. Ann. Inst. Fourier (Grenoble) 20 335–402. MR0283496 10.5802/aif.357
Boucheron, S., Lugosi, G. and Massart, P. (2000). A sharp concentration inequality with applications in random combinatorics and learning. Random Structures Algorithms 16 277–292. MR1749290 10.1002/(SICI)1098-2418(200005)16:3<277::AID-RSA4>3.0.CO;2-1 Boucheron, S., Lugosi, G. and Massart, P. (2000). A sharp concentration inequality with applications in random combinatorics and learning. Random Structures Algorithms 16 277–292. MR1749290 10.1002/(SICI)1098-2418(200005)16:3<277::AID-RSA4>3.0.CO;2-1
Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583–1614. MR1989444 1051.60020 10.1214/aop/1055425791 euclid.aop/1055425791 Boucheron, S., Lugosi, G. and Massart, P. (2003). Concentration inequalities using the entropy method. Ann. Probab. 31 1583–1614. MR1989444 1051.60020 10.1214/aop/1055425791 euclid.aop/1055425791
Devroye, L. (2002). Laws of large numbers and tail inequalities for random tries and Patricia trees. J. Comput. Appl. Math. 142 27–37. MR1910516 1005.60032 10.1016/S0377-0427(01)00458-7 Devroye, L. (2002). Laws of large numbers and tail inequalities for random tries and Patricia trees. J. Comput. Appl. Math. 142 27–37. MR1910516 1005.60032 10.1016/S0377-0427(01)00458-7
Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750. MR1410112 0867.60043 10.1214/aoap/1034968224 euclid.aoap/1034968224 Diaconis, P. and Saloff-Coste, L. (1996). Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695–750. MR1410112 0867.60043 10.1214/aoap/1034968224 euclid.aoap/1034968224
Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596. MR0615434 0481.62035 10.1214/aos/1176345462 euclid.aos/1176345462 Efron, B. and Stein, C. (1981). The jackknife estimate of variance. Ann. Statist. 9 586–596. MR0615434 0481.62035 10.1214/aos/1176345462 euclid.aos/1176345462
Frieze, A. (1985). On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56. MR0770868 10.1016/0166-218X(85)90058-7 Frieze, A. (1985). On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 47–56. MR0770868 10.1016/0166-218X(85)90058-7
Füredi, Z. and Komlós, J. (1981). The eigenvalues of random symmetric matrices. Combinatorica 1 233–241. MR0637828 10.1007/BF02579329 Füredi, Z. and Komlós, J. (1981). The eigenvalues of random symmetric matrices. Combinatorica 1 233–241. MR0637828 10.1007/BF02579329
Gross, L. (1993). Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet Forums (Varenna, 1992). Lecture Notes Math. 1563 54–88. Springer, Berlin. MR1292277 10.1007/BFb0074088 Gross, L. (1993). Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet Forums (Varenna, 1992). Lecture Notes Math. 1563 54–88. Springer, Berlin. MR1292277 10.1007/BFb0074088
Its, A. R., Tracy, C. A. and Widom, H. (2001). Random words, Toeplitz determinants and integrable systems I. In Random Matrix Models and Their Applications (P. Bleher and A. R. Its, eds.) 245–258. Math. Sci. Res. Inst. Publ. 40. Cambridge Univ. Press. MR1842789 0986.68104 Its, A. R., Tracy, C. A. and Widom, H. (2001). Random words, Toeplitz determinants and integrable systems I. In Random Matrix Models and Their Applications (P. Bleher and A. R. Its, eds.) 245–258. Math. Sci. Res. Inst. Publ. 40. Cambridge Univ. Press. MR1842789 0986.68104
Janson, S. (1995). The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Random Structures Algorithms 7 337–355. MR1369071 0844.05085 10.1002/rsa.3240070406 Janson, S. (1995). The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Random Structures Algorithms 7 337–355. MR1369071 0844.05085 10.1002/rsa.3240070406
Johansson, K. (2001). Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153 259–296. MR1826414 0984.15020 10.2307/2661375 Johansson, K. (2001). Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. Math. 153 259–296. MR1826414 0984.15020 10.2307/2661375
Ledoux, M. (1997). On Talagrand's deviation inequalities for product measures. ESAIM Probab. Math. Statist. 1 63–87. MR1399224 0869.60013 10.1051/ps:1997103 Ledoux, M. (1997). On Talagrand's deviation inequalities for product measures. ESAIM Probab. Math. Statist. 1 63–87. MR1399224 0869.60013 10.1051/ps:1997103
Ledoux, M. (2001). The Concentration of Measure Phenomenon. Amer. Math. Soc., Providence, RI. MR1849347 0995.60002 Ledoux, M. (2001). The Concentration of Measure Phenomenon. Amer. Math. Soc., Providence, RI. MR1849347 0995.60002
Linusson, S. and Wästlund, J. (2004). A proof of Parisi's conjecture on the random assignment problem. Probab. Theory Related Fields 128 419–440. MR2036492 10.1007/s00440-003-0308-9 Linusson, S. and Wästlund, J. (2004). A proof of Parisi's conjecture on the random assignment problem. Probab. Theory Related Fields 128 419–440. MR2036492 10.1007/s00440-003-0308-9
Maurer, A. (2006). Concentration inequalities for functions of independent variables. Random Structures Algorithms 29 121–138. MR2245497 10.1002/rsa.20105 Maurer, A. (2006). Concentration inequalities for functions of independent variables. Random Structures Algorithms 29 121–138. MR2245497 10.1002/rsa.20105
McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics (M. Habib, C. McDiarmid, J. Ramirez-Alfonsin and B. Reed, eds.) 195–248. Springer, New York. MR1678578 0927.60027 10.1007/978-3-662-12788-9 McDiarmid, C. (1998). Concentration. In Probabilistic Methods for Algorithmic Discrete Mathematics (M. Habib, C. McDiarmid, J. Ramirez-Alfonsin and B. Reed, eds.) 195–248. Springer, New York. MR1678578 0927.60027 10.1007/978-3-662-12788-9
McDiarmid, C. and Reed, B. (2006). Concentration of self-bounding functions and an inequality of Talagrand. Random Structures Algorithms 29 549–557. MR2268235 1120.60015 10.1002/rsa.20145 McDiarmid, C. and Reed, B. (2006). Concentration of self-bounding functions and an inequality of Talagrand. Random Structures Algorithms 29 549–557. MR2268235 1120.60015 10.1002/rsa.20145
Nair, C., Prabhakar, B. and Sharma, M. (2005). Proofs of the Parisi and Coppersmith–Sorkin random assignment conjectures. Random Structures Algorithms 27 413–444. MR2178256 1125.90026 10.1002/rsa.20084 Nair, C., Prabhakar, B. and Sharma, M. (2005). Proofs of the Parisi and Coppersmith–Sorkin random assignment conjectures. Random Structures Algorithms 27 413–444. MR2178256 1125.90026 10.1002/rsa.20084
Palmer, E. M. (1985). Graphical Evolution. Wiley, Chichester. MR0795795 0566.05002 Palmer, E. M. (1985). Graphical Evolution. Wiley, Chichester. MR0795795 0566.05002
Rhee, W.-S. and Talagrand, M. (1986). Martingale inequalities and the jackknife estimate of variance. Statist. Probab. Lett. 4 5–6. MR0822716 0593.62035 10.1016/0167-7152(86)90029-5 Rhee, W.-S. and Talagrand, M. (1986). Martingale inequalities and the jackknife estimate of variance. Statist. Probab. Lett. 4 5–6. MR0822716 0593.62035 10.1016/0167-7152(86)90029-5
Steele, J. M. (1986). An Efron–Stein inequality for nonsymmetric statistics. Ann. Statist. 14 753–758. MR0840528 0604.62017 10.1214/aos/1176349952 euclid.aos/1176349952 Steele, J. M. (1986). An Efron–Stein inequality for nonsymmetric statistics. Ann. Statist. 14 753–758. MR0840528 0604.62017 10.1214/aos/1176349952 euclid.aos/1176349952
Talagrand, M. (1994). On Russo's approximate zero–one law. Ann. Probab. 22 1576–1587. MR1303654 0819.28002 10.1214/aop/1176988612 euclid.aop/1176988612 Talagrand, M. (1994). On Russo's approximate zero–one law. Ann. Probab. 22 1576–1587. MR1303654 0819.28002 10.1214/aop/1176988612 euclid.aop/1176988612
Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. de Inst. Hautes Études Sci. 81 73–205. MR1361756 10.1007/BF02699376 Talagrand, M. (1995). Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. de Inst. Hautes Études Sci. 81 73–205. MR1361756 10.1007/BF02699376
Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563. MR1419006 0893.60001 10.1007/s002220050108 Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126 505–563. MR1419006 0893.60001 10.1007/s002220050108
Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1–34. MR1387624 10.1214/aop/1065725175 euclid.aop/1042644705 Talagrand, M. (1996). A new look at independence. Ann. Probab. 24 1–34. MR1387624 10.1214/aop/1065725175 euclid.aop/1042644705
Tracy, C. A. and Widom, H. (2001). On the distributions of the lengths of the longest monotone subsequences in random words. Probab. Theory Related Fields 119 350–380. MR1821139 10.1007/PL00008763 Tracy, C. A. and Widom, H. (2001). On the distributions of the lengths of the longest monotone subsequences in random words. Probab. Theory Related Fields 119 350–380. MR1821139 10.1007/PL00008763
Wästlund, J. (2005). A simple proof of the Parisi and Coppersmith–Sorkin formulas for the random assignment problem. Linköping Studies in Mathematics 6. Wästlund, J. (2005). A simple proof of the Parisi and Coppersmith–Sorkin formulas for the random assignment problem. Linköping Studies in Mathematics 6.
Wästlund, J. (2005). Evaluation of Janson's constant for the variance in the random minimum spanning tree problem. Linköping Studies in Mathematics 7. Wästlund, J. (2005). Evaluation of Janson's constant for the variance in the random minimum spanning tree problem. Linköping Studies in Mathematics 7.