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The arc-sine law and its analogs for processes governed by signed and complex measures

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  • Hochberg, Kenneth J.
  • Orsingher, Enzo

Abstract

The question whether the classical arc-sine law of Paul Lévy for the proportion of time spent by a Brownian particle on the positive half-line can be extended to generalized higher-order processes governed by signed and complex measures is studied. Both even and odd-order processes are considered, corresponding to heat-type equations with both real and imaginary coefficients. Finally, several mixtures of the earlier cases are analyzed as well.

Suggested Citation

  • Hochberg, Kenneth J. & Orsingher, Enzo, 1994. "The arc-sine law and its analogs for processes governed by signed and complex measures," Stochastic Processes and their Applications, Elsevier, vol. 52(2), pages 273-292, August.
    • Handle: RePEc:eee:spapps:v:52:y:1994:i:2:p:273-292
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    Citations

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    Cited by:

    1. Aimé Lachal, 2012. "A Survey on the Pseudo-process Driven by the High-order Heat-type Equation $\boldsymbol{\partial/\partial t=\pm\partial^N\!/\partial x^N}$ Concerning the Hitting and Sojourn Times," Methodology and Computing in Applied Probability, Springer, vol. 14(3), pages 549-566, September.
    2. Beghin, L. & Orsingher, E., 2005. "The distribution of the local time for "pseudoprocesses" and its connection with fractional diffusion equations," Stochastic Processes and their Applications, Elsevier, vol. 115(6), pages 1017-1040, June.
    3. Y. Nikitin & E. Orsingher, 2000. "On Sojourn Distributions of Processes Related to Some Higher-Order Heat-Type Equations," Journal of Theoretical Probability, Springer, vol. 13(4), pages 997-1012, October.
    4. Lachal, Aimé, 2014. "First exit time from a bounded interval for pseudo-processes driven by the equation ∂/∂t=(−1)N−1∂2N/∂x2N," Stochastic Processes and their Applications, Elsevier, vol. 124(2), pages 1084-1111.
    5. Beghin, L. & Orsingher, E. & Ragozina, T., 2001. "Joint distributions of the maximum and the process for higher-order diffusions," Stochastic Processes and their Applications, Elsevier, vol. 94(1), pages 71-93, July.
    6. Beghin, Luisa & Hochberg, Kenneth J. & Orsingher, Enzo, 2000. "Conditional maximal distributions of processes related to higher-order heat-type equations," Stochastic Processes and their Applications, Elsevier, vol. 85(2), pages 209-223, February.
    7. Bonaccorsi, Stefano & Calcaterra, Craig & Mazzucchi, Sonia, 2017. "An Itô calculus for a class of limit processes arising from random walks on the complex plane," Stochastic Processes and their Applications, Elsevier, vol. 127(9), pages 2816-2840.
    8. D’Ovidio, Mirko, 2011. "On the fractional counterpart of the higher-order equations," Statistics & Probability Letters, Elsevier, vol. 81(12), pages 1929-1939.
    9. Bonaccorsi, Stefano & Mazzucchi, Sonia, 2015. "High order heat-type equations and random walks on the complex plane," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 797-818.

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