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Dudley's theorem

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406: 219: 276: 546: 513: 116: 401:{\displaystyle \mathbf {E} \left\leq 24\int _{0}^{+\infty }{\sqrt {\log N(T,d_{X};\varepsilon )}}\,\mathrm {d} \varepsilon .} 35: 449: 103: 20: 453: 509: 471:(1967). "The sizes of compact subsets of Hilbert space and continuity of Gaussian processes". 468: 445: 59: 541: 501: 480: 39: 523: 492: 519: 488: 63: 250: 28: 58:
The result was first stated and proved by V. N. Sudakov, as pointed out in a paper by
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has a version with almost all sample path bounded and (uniformly) continuous on (
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Furthermore, if the entropy integral on the right-hand side converges, then
43: 458:. High Dimensional Probability. Vol. VII. pp. 37–43. 455:
V. N. Sudakov's work on expected suprema of Gaussian processes
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with making the connection between entropy and regularity.
214:{\displaystyle d_{X}(s,t)={\sqrt {\mathbf {E} {\big }}.\,} 279: 119: 400: 213: 291: 508:. Berlin: Springer-Verlag. pp. xii+480. 157: 8: 448:(2016). Houdré, Christian; Mason, David; 387: 386: 369: 345: 336: 331: 310: 294: 280: 278: 253:, i.e. the minimal number of (open) 210: 196: 191: 184: 171: 162: 156: 155: 150: 148: 124: 118: 547:Theorems regarding stochastic processes 437: 7: 388: 340: 14: 281: 151: 473:Journal of Functional Analysis 381: 356: 202: 192: 163: 142: 130: 93:be a Gaussian process and let 62:. Dudley had earlier credited 1: 228: > 0, denote by 16:Concept in probability theory 506:Probability in Banach spaces 485:10.1016/0022-1236(67)90017-1 452:; Jan Rosiński, Jan (eds.). 563: 27:is a result relating the 450:Reynaud-Bouret, Patricia 402: 215: 403: 216: 36:regularity properties 277: 117: 344: 469:Dudley, Richard M. 398: 327: 305: 266:required to cover 211: 21:probability theory 502:Talagrand, Michel 384: 290: 262:-balls of radius 205: 60:Richard M. Dudley 554: 528:(See chapter 11) 527: 500:Ledoux, Michel; 496: 460: 459: 442: 407: 405: 404: 399: 391: 385: 374: 373: 346: 343: 335: 320: 316: 315: 314: 304: 284: 220: 218: 217: 212: 206: 201: 200: 195: 189: 188: 176: 175: 166: 161: 160: 154: 149: 129: 128: 40:Gaussian process 25:Dudley's theorem 562: 561: 557: 556: 555: 553: 552: 551: 532: 531: 516: 499: 467: 464: 463: 446:Dudley, Richard 444: 443: 439: 434: 427: 365: 306: 289: 285: 275: 274: 261: 244: 190: 180: 167: 120: 115: 114: 101: 92: 82: 72: 64:Volker Strassen 56: 17: 12: 11: 5: 560: 558: 550: 549: 544: 534: 533: 530: 529: 514: 497: 479:(3): 290–330. 462: 461: 436: 435: 433: 430: 423: 409: 408: 397: 394: 390: 383: 380: 377: 372: 368: 364: 361: 358: 355: 352: 349: 342: 339: 334: 330: 326: 323: 319: 313: 309: 303: 300: 297: 293: 288: 283: 257: 251:entropy number 240: 222: 221: 209: 204: 199: 194: 187: 183: 179: 174: 170: 165: 159: 153: 147: 144: 141: 138: 135: 132: 127: 123: 97: 84: 78: 71: 68: 55: 52: 15: 13: 10: 9: 6: 4: 3: 2: 559: 548: 545: 543: 540: 539: 537: 525: 521: 517: 515:3-540-52013-9 511: 507: 503: 498: 494: 490: 486: 482: 478: 474: 470: 466: 465: 457: 456: 451: 447: 441: 438: 431: 429: 426: 422: 418: 414: 395: 392: 378: 375: 370: 366: 362: 359: 353: 350: 347: 337: 332: 328: 324: 321: 317: 311: 307: 301: 298: 295: 286: 273: 272: 271: 269: 265: 260: 256: 252: 248: 243: 239: 235: 231: 227: 207: 197: 185: 181: 177: 172: 168: 145: 139: 136: 133: 125: 121: 113: 112: 111: 109: 105: 100: 96: 91: 87: 81: 77: 69: 67: 65: 61: 53: 51: 49: 45: 41: 37: 33: 30: 26: 22: 505: 476: 472: 454: 440: 424: 420: 416: 412: 410: 267: 263: 258: 254: 246: 241: 237: 233: 229: 225: 223: 107: 104:pseudometric 98: 94: 89: 85: 79: 75: 73: 57: 24: 18: 110:defined by 50:structure. 32:upper bound 536:Categories 432:References 48:covariance 393:ε 379:ε 351:⁡ 341:∞ 329:∫ 322:≤ 299:∈ 178:− 70:Statement 504:(1991). 270:. Then 29:expected 542:Entropy 524:1102015 493:0220340 419:,  245:;  236:,  102:be the 54:History 44:entropy 42:to its 522:  512:  491:  249:) the 74:Let ( 38:of a 510:ISBN 224:For 46:and 34:and 481:doi 428:). 348:log 292:sup 106:on 19:In 538:: 520:MR 518:. 489:MR 487:. 475:. 325:24 23:, 526:. 495:. 483:: 477:1 425:X 421:d 417:T 413:X 396:. 389:d 382:) 376:; 371:X 367:d 363:, 360:T 357:( 354:N 338:+ 333:0 318:] 312:t 308:X 302:T 296:t 287:[ 282:E 268:T 264:ε 259:X 255:d 247:ε 242:X 238:d 234:T 232:( 230:N 226:ε 208:. 203:] 198:2 193:| 186:t 182:X 173:s 169:X 164:| 158:[ 152:E 146:= 143:) 140:t 137:, 134:s 131:( 126:X 122:d 108:T 99:X 95:d 90:T 88:∈ 86:t 83:) 80:t 76:X

Index

probability theory
expected
upper bound
regularity properties
Gaussian process
entropy
covariance
Richard M. Dudley
Volker Strassen
pseudometric
entropy number
Dudley, Richard
Reynaud-Bouret, Patricia
V. N. Sudakov's work on expected suprema of Gaussian processes
Dudley, Richard M.
doi
10.1016/0022-1236(67)90017-1
MR
0220340
Talagrand, Michel
ISBN
3-540-52013-9
MR
1102015
Categories
Entropy
Theorems regarding stochastic processes

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