Positivity of stable densities
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- by S. C. Port and R. A. Vitale PDF
- Proc. Amer. Math. Soc. 102 (1988), 1018-1023 Request permission
Abstract:
We settle two conjectures of Taylor about the positivity of the densities $p(t,x)$ of a drift-free, nondegenerate, stable process on $d$-dimensional Euclidean space ${R^d}$ starting at the origin. If $0 < \alpha < 1$ and $p(1,0) = 0$, we show that $x:\;p(t,x) > 0$ for some $t > 0$ is an open convex cone with vertex 0 and that $p(t,x) > 0$ for all $t > 0$ for each $x$ in this cone. If $\alpha = 1$ we show that $p(t,x) > 0$ for all $t > 0$ and all $x \in {R^d}$.References
- Sidney C. Port, A remark on hitting places for transient stable process, Ann. Math. Statist. 39 (1968), 365–371. MR 225386, DOI 10.1214/aoms/1177698397
- S. J. Taylor, Sample path properties of a transient stable process, J. Math. Mech. 16 (1967), 1229–1246. MR 0208684
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 1018-1023
- MSC: Primary 60E07; Secondary 60J45
- DOI: https://doi.org/10.1090/S0002-9939-1988-0934885-1
- MathSciNet review: 934885