| Preface |
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xiii | |
| Abbreviations |
|
xvii | |
| Notation |
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xix | |
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Stable random variables on the real line |
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1 | (54) |
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Equivalent definitions of a stable distribution |
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2 | (8) |
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Properties of stable random variables |
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10 | (10) |
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Symmetric α-stable random variables |
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20 | (1) |
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21 | (9) |
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Series representation of skewed α-stable random variables |
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30 | (5) |
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Graphs and tables of α-stable densities and c.d.f.'s |
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35 | (6) |
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41 | (8) |
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49 | (6) |
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Multivariate stable distributions |
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55 | (56) |
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57 | (6) |
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A counterexample for 0 < α < 1 |
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63 | (2) |
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Characteristic function of an α-stable random vector |
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65 | (7) |
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Strictly stable and symmetric stable random vectors |
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72 | (5) |
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Sub-Gaussian random vectors |
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77 | (7) |
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Complex SαS random variables |
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84 | (3) |
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87 | (8) |
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95 | (2) |
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97 | (6) |
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103 | (4) |
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107 | (4) |
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Stable random processes and stochastic integrals |
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111 | (62) |
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Stable stochastic processes |
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112 | (1) |
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Definition of stable integrals as a stochastic process |
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113 | (5) |
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118 | (3) |
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Constructive definition of stable integrals |
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121 | (5) |
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Properties of stable integrals |
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126 | (9) |
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135 | (7) |
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135 | (3) |
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138 | (1) |
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Ornstein-Uhlenbeck process |
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138 | (1) |
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Reverse Ornstein-Uhlenbeck process |
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139 | (1) |
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Well-balanced linear fractional stable motion |
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140 | (1) |
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Log-fractional stable motion |
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141 | (1) |
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Real stationary SαS harmonizable process |
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141 | (1) |
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142 | (1) |
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143 | (2) |
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Series representation for α-stable random measures |
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145 | (4) |
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A third definition of stable stochastic integrals using the series representation |
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149 | (3) |
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152 | (3) |
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A fourth definition of stable stochastic integrals using a Poisson representation |
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155 | (12) |
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167 | (6) |
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Dependence structures of multivariate stable distributions |
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173 | (50) |
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174 | (7) |
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Conditional laws that are symmetric around the conditional mean |
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181 | (4) |
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185 | (2) |
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Probability tails of order statistics |
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187 | (13) |
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200 | (4) |
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Association of stable random variables |
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204 | (4) |
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The codifference for stationary SαS processes |
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208 | (7) |
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The expected number of level crossings for stationary sub Gaussian processes |
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215 | (2) |
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217 | (6) |
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223 | (48) |
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Conditional moments of order greater than or equal to α |
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224 | (12) |
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Analytic representations of the non-linear regression functions |
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236 | (15) |
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251 | (4) |
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Graphical representations |
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255 | (5) |
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260 | (10) |
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270 | (1) |
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Complex stable stochastic integrals and harmonizable processes |
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271 | (38) |
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Complex-valued SαS random measures |
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272 | (3) |
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Integrals with respect to complex-valued SαS random measures |
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275 | (6) |
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The complex isotropic SαS case |
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281 | (5) |
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Series representation of complex-valued SαS random measures and integrals |
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286 | (5) |
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291 | (9) |
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Stationary real harmonizable processes |
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300 | (5) |
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The codifference for stationary real harmonizable processes |
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305 | (1) |
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306 | (3) |
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309 | (82) |
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311 | (7) |
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Fractional Brownian motion |
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318 | (22) |
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``Moving average'' representations of fractional Brownian motion |
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320 | (5) |
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``Harmonizable'' representations of fractional Brownian motion |
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325 | (7) |
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Fractional Gaussian noise |
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332 | (8) |
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General characteristics of processes that are α-stable and H-sssi |
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340 | (3) |
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Linear fractional stable motion |
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343 | (6) |
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349 | (3) |
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Log-fractional stable motion |
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352 | (3) |
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The real harmonizable fractional stable motion |
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355 | (3) |
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Complex harmonizable fractional stable motion |
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358 | (5) |
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363 | (3) |
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366 | (4) |
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Simulation of fractional noises and motions |
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370 | (6) |
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ARMA sequences with stable innovations |
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376 | (4) |
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Fractional ARIMA with stable innovations |
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380 | (7) |
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387 | (4) |
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391 | (28) |
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Self-similar fields with stationary increments in the strong sense |
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392 | (2) |
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394 | (6) |
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Example: the Levy-Chentsov random field |
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400 | (2) |
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Example: Takenaka random fields |
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402 | (3) |
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Properties of Chentsov random fields |
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405 | (2) |
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Properties of H-sssis Chentsov random fields |
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407 | (3) |
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Codifference induced by (α, H)-Takenaka fields |
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410 | (4) |
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Takenaka processes on (0, ∞) |
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414 | (3) |
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417 | (2) |
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Introduction to sample path properties |
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419 | (26) |
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420 | (1) |
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421 | (6) |
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427 | (3) |
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430 | (4) |
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434 | (5) |
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439 | (6) |
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Boundedness, continuity and oscillations |
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445 | (52) |
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446 | (1) |
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Necessary conditions for sample boundedness |
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447 | (8) |
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Necessary conditions for sample continuity |
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455 | (5) |
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Necessary and sufficient conditions for sample boundedness and continuity when 0 < α < 1 |
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460 | (10) |
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Probability tails of suprema of bounded α-stable processes, with index 0 < α < 2 |
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470 | (6) |
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476 | (6) |
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482 | (1) |
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483 | (1) |
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The level sets of the oscillation function |
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484 | (2) |
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A sample path alternative |
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486 | (2) |
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How strong is the basic assumption? |
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488 | (2) |
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490 | (7) |
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Measurability, integrability and absolute continuity |
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497 | (40) |
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Existence of a measurable version |
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498 | (4) |
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Integrability of the sample paths of stable processes |
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502 | (2) |
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Conditions for integrability |
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504 | (7) |
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Changing the order of integration |
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511 | (4) |
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Tail behavior of the LP-norm distribution |
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515 | (4) |
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Inversion formula for harmonizable SαS processes |
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519 | (5) |
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Absolute continuity of stable processes |
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524 | (9) |
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533 | (4) |
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Boundedness and continuity via metric entropy |
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537 | (22) |
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538 | (4) |
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Sufficient conditions in the case 1 ≤ α 7lt; 2 |
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542 | (4) |
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Necessary conditions in the case 1 ≤ α 7lt; 2 |
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546 | (4) |
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Boundedness and continuity of self-similar α-stable processes |
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550 | (6) |
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556 | (3) |
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559 | (12) |
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Countable parameter space |
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560 | (8) |
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Arbitrary parameter space |
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568 | (3) |
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Historical notes and extensions |
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571 | (26) |
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571 | (4) |
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575 | (2) |
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577 | (1) |
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578 | (4) |
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582 | (3) |
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585 | (1) |
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586 | (4) |
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590 | (2) |
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592 | (1) |
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592 | (1) |
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593 | (1) |
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594 | (1) |
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595 | (2) |
| Appendix: |
|
597 | (1) |
| A Table of symmetric α-stable fractiles |
|
597 | (6) |
| Bibliography |
|
603 | (18) |
| Subject index |
|
621 | (8) |
| Author index |
|
629 | |
Lisainfo e-raamatute kohta