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Substable and pseudo-isotropic processes. Connections with the geometry of subspaces of $L_α$

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Misiewicz J. K.

Instytut Matematyczny Polskiej Akademii Nauk

CONTENTS I. Introduction..........................................................................................................5 II. Pseudo-isotropic random vectors........................................................................9 II.1. Symmetric stable vectors................................................................................9 II.2. Pseudo-isotropic random vectors..................................................................15 II.3. Elliptically contoured vectors..........................................................................23 II.4. α-symmetric random vectors..........................................................................27 II.5. Substable random vectors.............................................................................32 III. Exchangeability and pseudo-isotropy.................................................................35 III.1. Pseudo-isotropic exchangeable sequences.................................................35 III.2. Schoenberg-type theorems..........................................................................40 III.3. Some generalizations...................................................................................43 IV. Stable and substable stochastic processes.....................................................45 IV.1. Gaussian processes and Reproducing Kernel Hilbert Spaces....................45 IV.2. Elliptically contoured processes..................................................................47 IV.3. Symmetric stable stochastic processes......................................................50 IV.4. Spectral representation of symmetric stable processes.............................56 IV.5. Substable and pseudo-isotropic stochastic processes...............................59 IV.6. $L_α $-dependent stochastic integrals.......................................................62 IV.7. Random limit theorems...............................................................................63 V. Infinite divisibility of substable stochastic processes..........................................64 V.1. Infinitely divisible distributions. Lévy measures............................................66 V.2. Approximative logarithm................................................................................68 V.3. Infinite divisibility of substable random vectors..............................................73 V.4. Infinite divisibility of substable processes......................................................77 References...........................................................................................................80 Index......................................................................................................................90

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1 Misiewicz J. K.

1 1996

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Substable and pseudo-isotropic processes. Connections with the geometry of subspaces of $L_α$