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Index

A a-equivalence, 74, 75 Ailam, G., 114 Aljancic, S., 25 Anderson, c., 94, 96, 97, 98, 106, 107 ARMA,225 Associated distribution, 176 Association, 298ff

and max-id random vectors, 299 and multivariate extreme value random

vectors, 299 and point proceses, 299, 300

Auxiliary function

B

for r-variation, 26, 27, 36, 40, 48, 64 for II-variation, 27, 30, 32, 35, 36 for Von Mises functions, 40, 64

Balkema, A., 35,43,53, 107,225,257, 262

Ballerini, R., 193, 194 Bamdorff-Nielsen, 0., 192 Billingsley, P., 125, 140, 150, 151, 152,

153, 196, 199,204,205,207, 217,221,230,231,238,269, 275,302

Bivariate densities Cauchy, 282, 287, 288 exponential, 289 F,286 gamma, 289 lognormal, 289 normal, 255ff, 297

t,286 Weibull, 106, 118

Boes, D., ix, 173 Breiman, L., 164 Burton, R., 299, 300, 305

C Cauchy distribution, 58, 63, 71, 108,

112, 121 bivariate, 282, 287, 288

Chung, K., 76, 80 Cinlar, E., 134 Cline, D., 59, 226 Cohen, J., 107 Cohn, D., 142 Continuous mapping theorem, 153 Convergence

almost sure, 6 continuous, I, 2 of densities, 85ff of moments, 76ff rates to extreme value laws, 107ff Skorohod metric, 196ff to types, 7ff uniform, 1-4, 17 vague, 139ff weak convergence of monotone

functions, 4, 5, 215 weak convergence of probability

measures or stochastic processes, 150ff, 204, 205, 209ff

Cramer-Wold device analogue for maxima, 275

316

D Davis, R .• 107, 226, 242, 276 Decoupage de Levy, 212, 215, 216, 244 D(O, 00), 196ff Deheuvels, P., 164. 195, 196, 243 Density convergence, 85ff Domain of attraction. 12, 28, 38ff

and asymptotic independence. 296 of A (x), 28, 38ff, 64, 66

representation, 43, 46, 64, 66 of multivariate extreme value distribu­

tion, 263. 276ff and multivariate regular variation,

277ff of <l>a (x), 15. 54ff, 63

representation, 58, 63 of'l'a (x), 59. 63

representation, 62. 63 of record value limit laws. 176ff

Durrett, R .• ~22. 223 Dwass. M .• 163. 169. 193. 194 Dynkin's Theorem. 125

E Equivalence classes (see n-varying func­

tions. r -varying functions; Tail equivalence; a-equivalence)

Esary. J .• 298, 299. 300 Excedances, 241. 242 Exponent measure. 259

and dependence, 294ff and independence, 29lff

Exponential distribution, 39, 42 bivariate, 289

Exponent of variation, 13 Extremal process, I 62ff, 179ff

asymptotic properties, 188, 189 construction using Poisson processes.

180 embedding a sequence of maxima in.

180 finite dimensional distribution. 179 inverse

has independent increments. 183 weak convergence to, 216ff. 240ff

jump times form a Poisson process, 184-186

Index

and Levy processes. 193 Markov property, 180 multivariate. 253 path properties. 180, 192 range is a Poisson process. 183 stochastic continuity. 180 weak limits of maxima, 209ff, 238ff

Extremes of moving averages. 238ff Extreme value distributions, 9ff

multivariate. 263ff

F F-density, 67, 75, 98, 121

multivariate, 286. 289 Feller, W .• 13, 14. 16.26.42,53,71,

170.227, 229, 255. 282 Finkelshteyn, B., 276 Finster. M., 226 Fisher, R., 107 Fixed discontinuities, 204 Functionals

G

continuity of, 148, 150, 161, 214 first jump bigger than (T2 ), 216 first jump past I. 219 first passage (T,), 217 jumps (T.), 218ff jump times past I (T,), 219-222, 224 largest jump (T,), 211, 213. 214. 238,

300 last jump before I, 219 linear combination. 227. 235 maximum modulus, 223 summation (T.). 223

Galambos. J .• 194. 273. 275, 279. 297 Gamma density, 72. 106. 121

bivariate. 289 r-varying functions. 26ff

and domains of attraction, 28 equivalence classes. 36 inversion of n-varying functions. 27 and Von Mises functions. 4Off, 66

Geffroy, J., 276, 297 Geometric distribution, 45 Gerritse, G., 257. 263

Gnedenko, 9, 16, 39, 45, 54, 59 Goldie, C., 13, 94, 97, 98, 164, 243,

248,249 Gumbel, E., vi Gumbel distribution, vi. 9ff, 38ff

H

Index

Haan, L. de, viii, 9, 13, 14, 16, 19,21, 22,28,30,35,36,37,43,48, ~,ll,64,~.~,~,~,M,

WI, 106, 107, 164,224,263, 264, 274, 275, 276. 284, 288. 290, 297, 298

Hall, J., WI, 114, 121 Hall, P. 107, 121 Hannan, T. 226 Hoeffding, W., 301 Hordijk, A., 37, 94, 101. 106 Husler, J., 224

I Ignatov, Z., 243 Intensity measure, 127 Invariance principle

applications, 215ff, 239ff Inverse of a monotone function, 3ff, 6

convergence of inverses, 5, 7, 8 relation between lI-varying and

r-varying functions, 27 Inverse process

J

independent increments, 167. 183, 193, 195, 196

weak convergence to, 215ff, 240ff

Jagers, P., 128, 141, 156,208 Johnson, N., 286, 289, 290

K Kallenberg, 0., 123, 141. 147, 150, 156,

161 Kanter, M., 226 Karamata, J., 25, 26 Karamata 's representation, 17

Karamata's Theorem, 17 Kendall's tau, 173 Kotz, S., 286, 289, 290 k-records, 242ff Kuratowski. I

L Lamperti, J., 163,211 Landau, E., 21 Laplace functional, 128ff, 140

of the Poisson process, 130ff

317

and weak convergence of point pro-cesses or random measures, 153ff

Large deviations for extremes, 94ff Leadbetter, R., viii, 156, 241 Lehmann, E., 300, 301 Levy, P., 212 Limit distributions

of maxima, 9ff of maxima and minima jointly, 239,

240, 242 of multivariate extremes, 264ff of sums, 12, 16

Lindgren, G., viii, 146, 241 Lindvall, T., 199,205,207,209 Loeve, M., 80 Logistic density, multivariate, 289 Lognormal density, 106

bivariate. 289

M Maller. R., 59 Mardia, K., 273, 276 Markov kernel, 164 Markov process, 164 Marshall, A., 273, 276, 289, 292, 297,

298,299 Maxima

comparison between continuous and discrete time, 191

embedding in an extremal process, 180 finite dimensional distributions of, 165 first passage process, 167 inverse process, 167 limit distributions of, 9ff Markov property, 165

318

Maxima (cont.)

of moving averages, 238 range is a Poisson process, 165-166 relative stability, 16, 53, 58, 179, 192 transition probabilities, 165 weak convergence to extremal

processes, 209ff Maximum modulus, terms of, 58 Max-infinitely divisible (max-id) distribu­

tions, 25 Iff and the bivariate normal distribution,

255ff characterization, 257ff, 260 closure under products, 253 closure under weak limits, 253 criteria, 254, 255, 257ff, 262 exponent measure, 259 and independence, 253, 259, 262,

291ff, 298 and multivariate extremal processes,

253ff, 260, 262-263 and Poisson processes, 258ff support of, 262 and weak limits, 262

Max self-decomposable, 263 Max-stable distributions, 264, 266ff

characterizations, 266ff and dependence, 294 and independence, 292, 296 with standardized marginals, 265

McCormick, W., 223 Mean measure, 127 Mejzler, D., 28 Miller, J., 194 Mises, R., Von, 21, 76 Moments, convergence of, 76ff Mori, T., 239 Moving averages, 224 Multivariate extremes, 250ff Multivariate extreme value distributions,

250ff and dependence, 294 domains of attraction, 276ff and independence, 292, 296 and max-stable' distributions, 264 representation, 268ff

Multivariate regular variation, 276ff and densities, 284ff and domains of attraction, 277ff

Index

N Neveu, J., 123, 149 Newman, C., 306 Normal distribution, 42, 71, 75, 121,

255f multivariate, 297

Normalizing constants, 67ff for attraction to A, 42, 48, 64 for attraction to <I> a' 15, 54, 63 for attraction to 'ITa, 59, 63 for attraction to record value limit

laws, 176, 177

o Olkin, 1.,273,276,289,292,297,298,

299 Omey, E., 284, 288, 290 One point uncompactification, 210, 225

p Pareto distribution, 58, 98 Pickands, J., 76, 84, 164,224,274,276 II-varying functions, 26ff

and derivatives, 30 and domains of attraction, 28 equivalence classes, 33 inverses of r-varying functions, 27 and - I variation, 30, 33 representations, 32 smooth versions, 33, 34

Point measures, 123 closed in the set of Radon measures,

145 mUltiplicity of, 124 Radon point measure, 124 relative compactness, criterion for,

146 simple, 124 support of, 124

Point processes, 123ff intensity measure, 127 Laplace functionals, 128ff mean measure, 127 and moving averages, 236 and rth largest order statistics, 239 simple, 156 weak convergence of, 150ff

Index

Point process method for proving weak convergence. 163. 209ff. 225ff

Poisson distribution. 45 Poisson process. 130ff

and association. 200 construction of. 1300f Laplace functional of, 130ff transformations of, 134ff

Pratt. J., 289 PRM (see Poisson process) Probability integral transform. 4 Proschan. F .. 298, 299. 300

R rth largest order statistic, 239. 242

convergence of. 239. 242 and point processes. 239

Radon point measure. 123 Random measures, 139ff Range of a sample. 251-252, 298 Rapidly varying functions. 16. 53. 67 Rates of convergence to extreme value

laws, 107ff Records. 162ff

k-records. 242ff limit laws, 174ff Markov property. 165ff of moving averages. 241 Poisson processes. 165ff relative stability. 178. 192 structure. 165ff transition probabilities. 165

Record times, 1 65ff, 169. 194. 195 asymptotic properties. 170. 189-191 comparison between discrete and

continuous time, 187ff independence property. 169 Poisson processes, 166. 170

Regularly varying functions multivariate case. 276ff. 284 univariate case, \3ff

Relative stability of maxima. 16,53.58,84. 179. 192 of records. 178. 192

Renyi, A .• 163, 169, 170 Resnick. S .• 35. 59. 74. 85, 94. 134.

161. 164, 192. 193. 194. 222, 223.224.226.227.242.248.

257. 262, 274. 275. 276. 284. 288, 290, 297, 298

Robbins. H .• 192 Rogers, C .• 243, 248. 249 Rootzen. H .. viii. 110, 122, 156. 241 Rubinovitch, M .. 164. 193

S Seneta, E., \3, 21, 25. 194 Shorrock. R., 164, 174, 195. 248 Sibuya. M .• 276. 297, 298 Siegmund. D .. 193 Simmons. G., 141, 146,267 Skorohod, A., 269 Skorohod's Theorem, 6. 7, 151 Skorohod topology. 196ff Slowly varying functions. 13ff Smirnov. N., 242 Smith, R., 94. 97, 98, 107, 112, 194 Stam. A .. 243. 288 Steiger. W .. 226 Stochastic continuity. 173. 204. 208 Stochastic ordering. 173 Stone. c., 199 Super slow variation. 97ff Sweeting. T .• 64. 85. 92

T Tail equivalence. 67ff I-density, 67, 75. 98. 121

bivariate. 286. 289 Teugels. J .• 13 Tiago de Oliviera. J .• 163. 276 Tightness, I 53ff, 161 Tippett. L.. 107 Tomkins. R .• 192 Tucker. H .• 26 Types. convergence to, 7ff

V Vague convergence. 142, 144 Vague topology. l40ff Vervaat. W., 164.243 Von Mises conditions. 62ff. 98, 104.

105. 107, 114

319

320

Yon Mises conditions (cont.) and local unifonn convergence of

densities, 85ff Yon Mises function, 40

W Walkup, D., 298, 299, 300 Waymire, E., 299, 300, 305 Weak convergence

of maxima, 2Wff, 238 of maxima and minima jointly, 239ff of moving averages, 236ff

Index

of point processes, 151ff to a Poisson process, 154

of probability measures or stochastic processes, 204, 205, 209ff, 216ff

of random measures, 151ff of sums, 222-223, 242

Weibull density, 106, 118 Weissman, I., 9, 12, 164 Wellner, J., WI, 114, 121 Welsch, R., 239 Westcott, M., 194 Whitt, W., 199, 205, 208, 213, 217, 221 Williams, D., 194 Wright, A., 306