Point-Processes

Book: Point-processes, Random measures, and stochastic geometry

F. Baccelli, B. Błaszczyszyn, M.K. Karray (2020). Random measures, point processes and stochastic geometry. [ HAL ]

This book is centered on the mathematical analysis of random structures embedded in the Euclidean space or more general topological spaces, with a main focus on random measures, point processes, and stochastic geometry. Such random structures have been known to play a key role in several branches of natural sciences (cosmology, ecology, cell biology) and engineering (material sciences, networks) for several decades. Their use is currently expanding to new fields like data sciences. The book was designed to help researchers finding a direct path from the basic definitions and properties of these mathematical objects to their use in new and concrete stochastic models.
The theory part of the book is structured to be self-contained, with all proofs included, in particular on measurability questions, and at the same time comprehensive. In addition to the illustrative examples which one finds in all classical mathematical books, the document features sections on more elaborate examples which are referred to as models in the book. Special care is taken to express these models, which stem from some of the natural sciences and engineering domains listed above, in clear and self-contained mathematical terms. This continuum from a comprehensive treatise on the theory of point processes and stochastic geometry to the collection of models that illustrate its representation power is probably the main originality of this book.
The book contains two types of mathematical results: (1) structural results on stationary random measures and stochastic geometry objects, which do not rely on any parametric assumptions; (2) more computational results on the most important parametric classes of point processes, in particular Poisson or Determinantal point processes. These two types are used to structure the book.
The material is organized as follows. Random measures and point processes are presented first, whereas stochastic geometry is discussed at the end of the book. For point processes and random measures, parametric models are discussed before non-parametric ones. For the stochastic geometry part, the objects of interest are often considered as point processes in the space of random sets of the Euclidean space. We discuss both general processes such as, e.g., particle processes, and parametric ones like, e.g., Poisson Boolean models of Poisson hyperplane processes.
We assume that the reader is acquainted with the basic results on measure and probability theories. We prove all technical auxiliary results when they are not easily available in the literature or when existing proofs appeared to us not sufficiently explicit. In all cases, the corresponding references will always be given.

Request for feedback

The present web version is a first version meant to trigger feedback. Comments from readers are most welcomed and should be sent to the authors at any of the following email addresses: francois.baccelli@ens.fr, bartek.blaszczyszyn@ens.fr, mohamed.karray@orange.com

Acknowledgements

The authors thank Oliver Diaz-Espinosa, Mayank Manjrekar, James Murphy, Eliza O’Reilly, and Pierre Bernhard for their comments on early versions of this manuscript.

Table of contents

I Random measures and point processes 1
 1 Foundations 3
  1.1 Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
  1.2 Mean measure, Laplace transform and void probability . . . . . . 6
   1.2.1 Campbell’s averaging formula . . . . . . . . . . . . . . . . 8
  1.3 Distribution characterization . . . . . . . . . . . . . . . . . . . . 9
   1.3.1 Powers and moment measures . . . . . . . . . . . . . . . . 11
   1.3.2 Laplace transform characterization . . . . . . . . . . . . . 12
   1.3.3 Independence . . . . . . . . . . . . . . . . . . . . . . . . . 12
  1.4 Stochastic integral . . . . . . . . . . . . . . . . . . . . . . . . . . 13
  1.5 Vague topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
  1.6 Point processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
   1.6.1 Simple point processes . . . . . . . . . . . . . . . . . . . . 16
   1.6.2 Enumeration of points . . . . . . . . . . . . . . . . . . . . 19
   1.6.3 Generating function . . . . . . . . . . . . . . . . . . . . . 24
   1.6.4 Factorial powers and moment measures . . . . . . . . . . 25
  1.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
 2 Basic models and operations 29
  2.1 Poisson point processes . . . . . . . . . . . . . . . . . . . . . . . 29
   2.1.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . 30
   2.1.2 Characterizations . . . . . . . . . . . . . . . . . . . . . . . 32
  2.2 Operations on rand. measures and point proc. . . . . . . . . . . . 35
   2.2.1 Superposition . . . . . . . . . . . . . . . . . . . . . . . . . 35
   2.2.2 Thinning of points . . . . . . . . . . . . . . . . . . . . . . 37
   2.2.3 Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
   2.2.4 Independent displacement of points . . . . . . . . . . . . . 41
   2.2.5 Independent marking of points . . . . . . . . . . . . . . . 43
   2.2.6 Marked random measures . . . . . . . . . . . . . . . . . . 45
   2.2.7 Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
 2.3 Constructing new models . . . . . . . . . . . . . . . . . . . . . . 49
   2.3.1 Cox point processes . . . . . . . . . . . . . . . . . . . . . 49
   2.3.2 Gibbs point processes . . . . . . . . . . . . . . . . . . . . 52
   2.3.3 Cluster point processes . . . . . . . . . . . . . . . . . . . 53
   2.3.4 Powers and factorial powers . . . . . . . . . . . . . . . . . 61
  2.4 Shot-noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
   2.4.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . 66
   2.4.2 Second order moments . . . . . . . . . . . . . . . . . . . . 68
   2.4.3 U-statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 74
  2.5 Sigma-finite random measures . . . . . . . . . . . . . . . . . . . . 75
  2.6 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
   2.6.1 For Section 2.1 . . . . . . . . . . . . . . . . . . . . . . . . 76
   2.6.2 For Section 2.2 . . . . . . . . . . . . . . . . . . . . . . . . 77
   2.6.3 For Section 2.4 . . . . . . . . . . . . . . . . . . . . . . . . 81
  2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
   2.7.1 For Section 2.1 . . . . . . . . . . . . . . . . . . . . . . . . 87
   2.7.2 For Section 2.2 . . . . . . . . . . . . . . . . . . . . . . . . 89
   2.7.3 For Section 2.3 . . . . . . . . . . . . . . . . . . . . . . . . 96
   2.7.4 For Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 98
 3 Palm theory 119
  3.1 Palm distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 119
   3.1.1 Reduced Palm distribution . . . . . . . . . . . . . . . . . 123
   3.1.2 Mixed Palm version . . . . . . . . . . . . . . . . . . . . . 124
   3.1.3 Local Palm probabilities . . . . . . . . . . . . . . . . . . . 126
  3.2 Palm distributions for particular models . . . . . . . . . . . . . . 127
   3.2.1 Palm for Poisson point processes . . . . . . . . . . . . . . 127
   3.2.2 Palm for Cox point processes . . . . . . . . . . . . . . . . 131
   3.2.3 Palm distribution of Gibbs point processes . . . . . . . . 132
   3.2.4 Palm distribution for marked random measures . . . . . . 136
  3.3 Higher order Palm and reduced Palm . . . . . . . . . . . . . . . . 141
   3.3.1 Higher order Palm . . . . . . . . . . . . . . . . . . . . . . 141
   3.3.2 Higher order reduced Palm . . . . . . . . . . . . . . . . . 143
  3.4 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
   3.4.1 For Section 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 147
   3.4.2 For Section 3.3 . . . . . . . . . . . . . . . . . . . . . . . . 151
  3.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
   3.5.1 For Section 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 153
 4 Transforms and moment measures 161
  4.1 Characteristic function . . . . . . . . . . . . . . . . . . . . . . . . 161
   4.1.1 Cumulant measures . . . . . . . . . . . . . . . . . . . . . 162
   4.1.2 Factorial cumulant measures . . . . . . . . . . . . . . . . 165
  4.2 Finite series transform expansions . . . . . . . . . . . . . . . . . 167
   4.2.1 Characteristic function expansion . . . . . . . . . . . . . . 167
   4.2.2 Laplace transform expansion . . . . . . . . . . . . . . . . 169
   4.2.3 Generating function expansion . . . . . . . . . . . . . . . 170
  4.3 Infinite series transform expansions . . . . . . . . . . . . . . . . . 172
   4.3.1 Void probability expansion . . . . . . . . . . . . . . . . . 172
   4.3.2 Symmetric enumeration of atoms of finite point processes 172
   4.3.3 Janossy measures . . . . . . . . . . . . . . . . . . . . . . . 174
   4.3.4 Moment versus Janossy measures . . . . . . . . . . . . . . 176
   4.3.5 Janossy versus moment measures . . . . . . . . . . . . . . 177
   4.3.6 Distribution of a finite point process . . . . . . . . . . . . 184
   4.3.7 Order statistics on R . . . . . . . . . . . . . . . . . . . . . 187
  4.4 Factorial moment expansion . . . . . . . . . . . . . . . . . . . . . 195
   4.4.1 Point processes on R . . . . . . . . . . . . . . . . . . . . . 195
   4.4.2 General marked point processes . . . . . . . . . . . . . . . 201
   4.4.3 Shot-noise functions . . . . . . . . . . . . . . . . . . . . . 210
  4.5 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
   4.5.1 For Section 4.2 . . . . . . . . . . . . . . . . . . . . . . . . 214
   4.5.2 For Section 4.4 . . . . . . . . . . . . . . . . . . . . . . . . 217
  4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
   4.6.1 For Section 4.1 . . . . . . . . . . . . . . . . . . . . . . . . 220
   4.6.2 For Section 4.2 . . . . . . . . . . . . . . . . . . . . . . . . 222
 5 Determinantal and permanental processes 227
  5.1 Determinantal point process basics . . . . . . . . . . . . . . . . . 227
   5.1.1 Definition and basic properties . . . . . . . . . . . . . . . 227
   5.1.2 Indistinguishable kernels . . . . . . . . . . . . . . . . . . . 230
   5.1.3 Uniqueness of the distribution . . . . . . . . . . . . . . . . 232
   5.1.4 Generating function and Laplace transform . . . . . . . . 234
   5.1.5 Inequalities for moment measures . . . . . . . . . . . . . . 235
  5.2 Existence of det. point proc. with regular kern. . . . . . . . . . . 236
   5.2.1 Canonical determinantal point processes . . . . . . . . . . 236
   5.2.2 Integral operator: essentials . . . . . . . . . . . . . . . . . 242
   5.2.3 Canonical version of a kernel . . . . . . . . . . . . . . . . 243
   5.2.4 Regular kernels . . . . . . . . . . . . . . . . . . . . . . . . 244
  5.3 α-Determinantal point processes . . . . . . . . . . . . . . . . . . 250
   5.3.1 Definition and basic properties . . . . . . . . . . . . . . . 250
   5.3.2 Uniqueness of distribution . . . . . . . . . . . . . . . . . . 253
   5.3.3 Generating function and Laplace transform . . . . . . . . 255
   5.3.4 Permanental point process as Cox point process . . . . . . 257
   5.3.5 Existence of α-determinantal point processes . . . . . . . 259
  5.4 Laplace transform and Janossy measures revisited . . . . . . . . 261
   5.4.1 Laplace transform as operator determinant . . . . . . . . 261
   5.4.2 Janossy measures of α-determinantal point processes; α ∈{1/m : m ∈ N∗} . . . . . . . . . . . . . . . . . . . . . . . . 263
   5.4.3 Janossy measures of α-determinantal point processes; α ∈
{−1/m : m ∈ N∗} . . . . . . . . . . . . . . . . . . . . . . . 270
  5.5 Palm distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 273
  5.6 Stationary determinantal point processes on Rd . . . . . . . . . . 275
   5.6.1 Ginibre determinantal point process . . . . . . . . . . . . 276
   5.6.2 Shift-invariant kernel . . . . . . . . . . . . . . . . . . . . . 279
  5.7 Discrete determinantal point processes . . . . . . . . . . . . . . . 284
   5.7.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . 285
   5.7.2 Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
   5.7.3 Janossy measures . . . . . . . . . . . . . . . . . . . . . . . 288
   5.7.4 Palm version . . . . . . . . . . . . . . . . . . . . . . . . . 289
  5.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
   5.8.1 For Section 5.1 . . . . . . . . . . . . . . . . . . . . . . . . 292
   5.8.2 For Section 5.2 . . . . . . . . . . . . . . . . . . . . . . . . 292
   5.8.3 For Section 5.3 . . . . . . . . . . . . . . . . . . . . . . . . 293
   5.8.4 For Section 5.6 . . . . . . . . . . . . . . . . . . . . . . . . 293
II Stationary random measures and point processes 297
 6 Palm theory in the stationary framework 299
  6.1 Palm probabilities in the stationary framework . . . . . . . . . . 299
   6.1.1 Stationary framework . . . . . . . . . . . . . . . . . . . . 299
   6.1.2 Palm probability of a random measure . . . . . . . . . . . 304
   6.1.3 Campbell-Little-Mecke-Matthes theorem . . . . . . . . . . 309
   6.1.4 Mass transport formula . . . . . . . . . . . . . . . . . . . 313
   6.1.5 Mecke’s invariance theorem . . . . . . . . . . . . . . . . . 317
  6.2 Palm inversion formula . . . . . . . . . . . . . . . . . . . . . . . . 320
   6.2.1 Voronoi tessellation . . . . . . . . . . . . . . . . . . . . . 320
   6.2.2 Inversion formula . . . . . . . . . . . . . . . . . . . . . . . 322
   6.2.3 Typical versus zero cell . . . . . . . . . . . . . . . . . . . 325
   6.2.4 Particular case of the line . . . . . . . . . . . . . . . . . . 329
   6.2.5 Renewal processes . . . . . . . . . . . . . . . . . . . . . . 331
   6.2.6 Direct and inverse construction of Palm theory . . . . . . 333
  6.3 Further properties of Palm probabilities . . . . . . . . . . . . . . 334
   6.3.1 Independence . . . . . . . . . . . . . . . . . . . . . . . . . 334
   6.3.2 Superposition . . . . . . . . . . . . . . . . . . . . . . . . . 337
   6.3.3 Neveu’s exchange formula . . . . . . . . . . . . . . . . . . 339
   6.3.4 Alternative version of Neveu’s exchange theorem . . . . . 345
   6.3.5 The Holroyd-Peres representation of Palm probability . . 347
   6.3.6 Reduced second moment measure . . . . . . . . . . . . . . 349
  6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
    6.4.1 For Section 6.1 . . . . . . . . . . . . . . . . . . . . . . . . 350
   6.4.2 For Section 6.2 . . . . . . . . . . . . . . . . . . . . . . . . 358
   6.4.3 For Section 6.3 . . . . . . . . . . . . . . . . . . . . . . . . 359
 7 Marks in the stationary framework 365
  7.1 Stationary marked random measures . . . . . . . . . . . . . . . . 365
   7.1.1 Stationary marked point processes . . . . . . . . . . . . . 367
   7.1.2 Extension of PASTA to Rd . . . . . . . . . . . . . . . . . 373
  7.2 Marks in a general measurable space . . . . . . . . . . . . . . . . 374
   7.2.1 Selected marks and conditioning . . . . . . . . . . . . . . 375
   7.2.2 Transformations of stationary point process based on marks377
  7.3 Palm theory for stationary marked random measures . . . . . . . 380
   7.3.1 Palm distribution of the mark . . . . . . . . . . . . . . . . 380
   7.3.2 Palm distributions of marked random measure . . . . . . 381
   7.3.3 Palm probability conditional on the mark . . . . . . . . . 382
  7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
   7.4.1 For Section 7.1.1 . . . . . . . . . . . . . . . . . . . . . . . 384
 8 Ergodicity 399
  8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399
  8.2 Birkhoff’s pointwise ergodic theorem . . . . . . . . . . . . . . . . 400
   8.2.1 Ergodic theory . . . . . . . . . . . . . . . . . . . . . . . . 400
  8.3 Ergodic theorems for random measures . . . . . . . . . . . . . . . 402
   8.3.1 Ergodicity of random measures . . . . . . . . . . . . . . . 402
   8.3.2 Ergodic theorem for random measures . . . . . . . . . . . 403
   8.3.3 Cross-ergodicity . . . . . . . . . . . . . . . . . . . . . . . 412
  8.4 Ergodicity of marked random measures . . . . . . . . . . . . . . 413
  8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
III Stochastic geometry 417
 9 Framework for stochastic geometry 419
  9.1 Space of closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . 419
  9.2 Random closed sets . . . . . . . . . . . . . . . . . . . . . . . . . . 425
   9.2.1 The capacity functional . . . . . . . . . . . . . . . . . . . 427
   9.2.2 Set processes . . . . . . . . . . . . . . . . . . . . . . . . . 430
   9.2.3 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . 431
   9.2.4 Characteristics of random closed set . . . . . . . . . . . . 431
   9.2.5 Characteristics of stationary random closed set . . . . . . 432
  9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
   9.3.1 For Section 9.2 . . . . . . . . . . . . . . . . . . . . . . . . 435
 10 Coverage and germ-grain models 439
  10.1 Coverage model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
  10.2 Germ-grain model . . . . . . . . . . . . . . . . . . . . . . . . . . 441
   10.2.1 Germ-grain construction . . . . . . . . . . . . . . . . . . . 441
   10.2.2 Inverse construction . . . . . . . . . . . . . . . . . . . . . 445
  10.3 Further examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
   10.3.1 Hard-core coverage models . . . . . . . . . . . . . . . . . 447
   10.3.2 Shot-noise coverage models . . . . . . . . . . . . . . . . . 448
  10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
 11 Line processes and tessellations 457
  11.1 Line processes in R2 . . . . . . . . . . . . . . . . . . . . . . . . . 457
   11.1.1 Parameterization of lines in R2 . . . . . . . . . . . . . . . 457
   11.1.2 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . 458
   11.1.3 Associated random measures . . . . . . . . . . . . . . . . 460
  11.2 Planar tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . 463
   11.2.1 Voronoi tessellation . . . . . . . . . . . . . . . . . . . . . 463
   11.2.2 Crofton tessellation . . . . . . . . . . . . . . . . . . . . . . 466
  11.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
 12 Complements 475
  12.1 Strong Markov property of Poisson point process . . . . . . . . . 475
IV Appendix 477
 13 Transforms of random variables 479
  13.A Random variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 479
   13.A.1 Characteristic function . . . . . . . . . . . . . . . . . . . . 479
   13.A.2 Generating function . . . . . . . . . . . . . . . . . . . . . 484
   13.A.3 Moments versus factorial moments . . . . . . . . . . . . . 488
   13.A.4 Ordinary cumulants . . . . . . . . . . . . . . . . . . . . . 491
   13.A.5 Factorial cumulants . . . . . . . . . . . . . . . . . . . . . 492
  13.B Nonnegative random variables . . . . . . . . . . . . . . . . . . . . 499
   13.B.1 Laplace transform . . . . . . . . . . . . . . . . . . . . . . 499
   13.B.2 Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . 503
  13.C Random vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504
   13.C.1 Moments from transforms . . . . . . . . . . . . . . . . . . 504
   13.C.2 Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . 505
   13.C.3 Nonnegative random vectors . . . . . . . . . . . . . . . . 509
 14 Useful results in measure theory 511
  14.A Basic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511
  14.B Support of a measure . . . . . . . . . . . . . . . . . . . . . . . . . 514
  14.C Functional monotone class theorems . . . . . . . . . . . . . . . . 516
  14.D Mixture and disintegration of measures . . . . . . . . . . . . . . . 517
   14.D.1 Mixture of measures . . . . . . . . . . . . . . . . . . . . . 517
   14.D.2 Disintegration of measures . . . . . . . . . . . . . . . . . . 520
  14.E Power and factorial powers of measures . . . . . . . . . . . . . . 527
  14.F Equality of measures . . . . . . . . . . . . . . . . . . . . . . . . . 531
  14.G Symmetric complex Gaussian random variables . . . . . . . . . . 532
 15 Useful results in algebra 535
   15.A Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
   15.A.1 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 535
   15.A.2 Schur complement . . . . . . . . . . . . . . . . . . . . . . 536
   15.A.3 Diagonal expansion of the determinant . . . . . . . . . . . 537
   15.A.4 α-determinant of a matrix . . . . . . . . . . . . . . . . . . 538
  15.B Power series composition . . . . . . . . . . . . . . . . . . . . . . . 539
 16 Useful results in functional analysis 541
  16.A Integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
   16.A.1 L2 space properties . . . . . . . . . . . . . . . . . . . . . . 541
   16.A.2 Linear operators . . . . . . . . . . . . . . . . . . . . . . . 543
   16.A.3 Integral operator basics . . . . . . . . . . . . . . . . . . . 544
   16.A.4 Trace cass operators . . . . . . . . . . . . . . . . . . . . . 555
   16.A.5 Hilbert-Schmidt operators . . . . . . . . . . . . . . . . . . 558
   16.A.6 Restriction property . . . . . . . . . . . . . . . . . . . . . 559
   16.A.7 Canonical and pre-canonical kernels . . . . . . . . . . . . 561
   16.A.8 Continuous kernels . . . . . . . . . . . . . . . . . . . . . . 567
   16.A.9 Nonnegative-definite property . . . . . . . . . . . . . . . . 570
  16.B Operator determinant . . . . . . . . . . . . . . . . . . . . . . . . 571
   16.B.1 Fredholm determinant . . . . . . . . . . . . . . . . . . . . 571
   16.B.2 Expansion of integral operator’s determinant . . . . . . . 576

Mohamed Kadhem KARRAY

My research activities at Orange aim to evaluate the performance of communication networks, by combining information, queueing theories, stochastic geometry, as well as machine and deep learning. Recently, I prepared video lectures on "Data science: From multivariate statistics to machine and deep learning" available on my YouTube channel. I also teached at Ecole Normale Supérieure, Ecole Polyetechnique, Ecole Centrale Paris, and prepared several mathematical books.

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