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On the $\gamma$Variation of Processes with Stationary Independent Increments

# On the $\gamma$Variation of Processes with Stationary Independent Increments,10.1214/aoms/1177692473,The Annals of Mathematical Statistics,Itrel Monro

On the $\gamma$Variation of Processes with Stationary Independent Increments
Let $\{X_t; t \geqq 0\}$ be a stochastic process in $R^N$ defined on the probability space $(\Omega, \mathscr{F}, \mathbf{P})$ which has stationary independent increments. Let $\nu$ be the Levy measure for $X_t$ and let $\beta = \inf\{\alpha > 0: \int_{|x| < 1}|x|^\alpha\nu(dx) < \infty\}$. For each $\omega \in \Omega$, let $V_\gamma(\mathbf{X}(\bullet, \omega); a, b) = \sup \sum^m_{j=1} |X(t_j, \omega) - X(_{t-1}, \omega)|^\gamma$ where the supremum is over all finite subdivisions $a = t_0 < t_1 < \cdots < t_m = b$. Then if $\gamma > \beta, \mathbf{P}\{\mathbf{V}_\gamma(\mathbf{X}(\bullet, \omega); a, b) < \infty\} = 1$.
Journal: The Annals of Mathematical Statistics , vol. 43, no. 1972, pp. 1213-1220, 1972
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## Citation Context (5)

• ...The results of Blumenthal and Getoor [5, Theorems 4.1, 4.2] and Monroe [23, Theorem 2] combine to provide a similar result for normalized Lévy processes...
• ...Theorem 6.4 [5, 23] Let X ={ Xt ,t ≥ 0} be a normalized Lévy process in R d . Then...

### Jamison Wolf. Random Fractals Determined by Lévy Processes

• ...Theorem 1.5 [15, Theorem 2] Let (Xt)t≥0 be a L´evy process in Rn without a...
• ...[15, Theorem 1] shows that the index of the process τ(s) is half that of the...

### David R. E. Williams. Path-wise solutions of SDE's driven by Levy processes

• ...Remark 4.1 In [13] the characterisation of the sample path p-variation of all L´evy processes is completed...

### David R. E. Williams. Diffeomorphic flows driven by Levy processes

• ...ments of a stochastically continuous process with stationary independent increments, the behavior of V~(g, S) has been extensively studied in [3, 8, 9, 11, 12, 16] and [17]...

### Patrick L. Brockett. Variational sums of infinitesimal systems

• ...Getoor [3] and Monroe [19] on "strong" variation generalize in an obvious way to ich...

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## Citations (15)

### Random Fractals Determined by Lévy Processes

Journal: Journal of Theoretical Probability - J THEOR PROBABILITY , vol. 23, no. 4, pp. 1182-1203, 2010

### First order p -variations and Besov spaces(Citations: 2)

Journal: Statistics & Probability Letters - STAT PROBAB LETT , vol. 79, no. 1, pp. 55-62, 2009

### Generalized fractional Ornstein-Uhlenbeck processes

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### Rough functions: p Variation, calculus, and index estimation(Citations: 2)

Journal: Lithuanian Mathematical Journal - LITH MATH J , vol. 46, no. 1, pp. 102-128, 2006

### Quadratic variation, p-variation and integration with applications to stock price modelling

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