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Preface | |

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Vector Integration | |

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Preliminaries | |

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Banach spaces | |

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Classes of sets | |

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Measurable functions | |

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Simple measurability of operator-valued functions | |

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Weak measurability | |

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Integral of step functions | |

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Totally measurable functions and the immediate integral | |

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The Riesz representation theorem | |

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The classical integral | |

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The Bochner integral | |

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Convergence theorems | |

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Measures with finite variation | |

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The variation of vector measures | |

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Boundedness of [sigma]-additive measures | |

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Variation of real-valued measures | |

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Integration with respect to vector measures with finite variation | |

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The indefinite integral | |

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Integration with respect to gm | |

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The Radon-Nikodym theorem | |

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Conditional expectations | |

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[sigma]-additive measures | |

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[sigma]-additive measures on [sigma]-rings | |

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Uniform [sigma]-additivity | |

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Uniform absolute continuity and uniform [sigma]-additivity | |

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Weak [sigma]-additivity | |

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Uniform [sigma]-additivity of indefinite integrals | |

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Weakly compact sets in L[superscript 1 subscript F] ([mu]) | |

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Measures with finite semivariation | |

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The semivariation | |

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Semivariation and norming spaces | |

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The semivariation of [sigma]-additive measures | |

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The family m[subscript F,Z] of measures | |

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Integration with respect to a measure with finite semivariation | |

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Measurability with respect to a vector measure | |

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The seminorm m[subscript F,G](f) | |

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The space of integrable functions | |

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The integral | |

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Convergence theorems | |

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Properties of the space F[subscript D] (B, m[subscript F,G]) | |

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Relationship between the spaces F[subscript D](m) | |

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The indefinite integral of measures with finite semivariation | |

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Strong additivity | |

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Extension of measures | |

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Applications | |

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The Riesz representation theorem | |

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Integral representation of continuous linear operations on L[superscript p]-spaces | |

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Random Gaussian measures | |

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The Stochastic Integral | |

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Summable processes | |

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Notations | |

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The measure I[subscript X] | |

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Summable processes | |

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Computation of I[subscript X] for predictable rectangles | |

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Computation of I[subscript X] for stochastic intervals | |

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The stochastic integral | |

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The space F[subscript D] [characters not reproducible] | |

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The integral [function of] HdI[subscript X] | |

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A convergence theorem | |

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The stochastic integral H - X | |

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The stochastic integral and stopping times | |

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Stochastic integral of elementary processes | |

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Stopping the stochastic integral | |

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Summability of stopped processes | |

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The jumps of the stochastic integral | |

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Convergence theorems | |

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The completeness of the space L[superscript 1 subscript F,G](X) | |

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The Uniform Convergence Theorem | |

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The Vitali and the Lebesgue Convergence Theorems | |

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The stochastic integral of [sigma]-elementary and of caglad processes as a pathwise Stieltjes integral | |

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Summability of the stochastic integral | |

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Summability criterion | |

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Quasimartingales and the Doleans measure | |

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The summability criterion | |

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Local summability and local integrability | |

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Definitions | |

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Basic properties | |

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Convergence theorems | |

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Additional properties | |

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Martingales | |

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Stochastic integral of martingales | |

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Square integrable martingales | |

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Extension of the measure I[subscript M] | |

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Summability of square integrable martingales | |

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Properties of the space F[subscript F,G](M) | |

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Isometrical isomorphism of L[superscript 1 subscript F,G](M) and L[superscript 2 subscript F]([mu subscript [M]]) | |

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Processes with Finite Variation | |

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Functions with finite variation and their Stieltjes integral | |

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Functions with finite variation | |

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The variation function g | |

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The measure associated to a function | |

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The Stieltjes integral | |

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Processes with finite variation | |

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Definition and properties | |

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Optional and predictable measures | |

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The measure [mu subscript X] | |

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Summability of processes with integrable variation | |

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The stochastic integral as a Stieltjes integral | |

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The pathwise stochastic integral | |

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Semilocally summable processes | |

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Processes with Finite Semivariation | |

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Functions with finite semivariation and their Stieltjes integral | |

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Functions with finite semivariation | |

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Semivariation and norming spaces | |

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The measure associated to a function | |

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The Stieltjes integral with respect to a function with finite semivariation | |

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Processes with finite semivariation | |

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The semivariation process | |

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The measure [mu]x | |

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Summability of processes with integrable semivariation | |

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The pathwise stochastic integral | |

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Dual projections | |

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Dual projection of measures | |

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Dual projections of processes | |

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Existence of dual projections | |

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Processes with locally integrable variation or semivariation | |

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Examples of processes with locally integrable variation or semivariation | |

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Decomposition of local martingales | |

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The Ito Formula | |

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The Ito formula | |

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Preliminary results | |

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The vector quadratic variation | |

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The quadratic variation | |

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The process of jumps | |

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Ito's formula | |

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Stochastic Integration in the Plane | |

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Preliminaries | |

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Order relation in R[superscript 2] | |

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The increment [Delta subscript zz], g | |

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Right continuity | |

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The filtration | |

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The predictable [Sigma]-algebra | |

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Stopping times | |

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Stochastic processes | |

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Extension of processes from R[superscript 2 subscript +] [times] [Omega] to R[superscript 2] [times] [Omega] | |

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Summable processes | |

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The measure I[subscript X] | |

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Summable processes | |

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The seminorm I[subscript X] and the space F[subscript F,G](X) | |

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The integral [function of] HdI[subscript X] | |

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The stochastic integral H - X | |

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Properties of the stochastic integral | |

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Convergence theorems | |

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Extension of I[subscript X] to P([infinity]) | |

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Existence of left limits of X in L[superscript p subscript E] | |

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Some properties of the integral [function of] HdI[subscript X] | |

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Summability of stopped processes | |

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Summability of the stochastic integral | |

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Two-Parameter Martingales | |

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Martingales | |

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Square integrable martingales | |

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A decomposition theorem | |

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The measures [characters not reproducible] and [mu subscript [M]] | |

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Summability of the square integrable martingales in Hilbert spaces | |

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The space F[subscript F,G](I[subscript M]) | |

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Isometric isomorphism of L[superscript 1 subscript F,G](M) and L[superscript 2 subscript F]([mu subscript [M]]) | |

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Two-Parameter Processes with Finite Variation | |

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Functions with finite variation in the plane | |

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Monotone functions | |

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Partitions | |

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Variation corresponding to a partition | |

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Variation of a function on a rectangle | |

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Limits of the variation | |

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The variation function g | |

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Functions with finite variation | |

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Functions vanishing outside a quadrant | |

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Variation of real-valued functions | |

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Lateral limits | |

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Measures associated to functions | |

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[sigma]-additivity of the measure m[subscript g] | |

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The Stieltjes integral | |

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Processes with finite variation | |

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Processes with integrable variation | |

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The measure [mu subscript X] | |

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Summability of processes with integrable variation | |

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The stochastic integral as a Stieltjes integral | |

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Two-Parameter Processes with Finite Semivariation | |

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Functions with finite semivariation in the plane | |

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Functions with finite semivariation | |

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Semivariation and norming spaces | |

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The measure associated to a function | |

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The Stieltjes integral for functions with finite semivariation in R[superscript 2] | |

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Processes with finite semivariation in the plane | |

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Processes with finite semivariation | |

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The measure [mu subscript X] | |

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Summability of processes with integrable semivariation | |

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References | |