ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary LetX be a diffusion in natural scale on (0,1], with 1 reflecting, and letc(x)≡ℍ(H x ) andv(x)≡var (H x ), whereH x =inf{t: X t =x}. Let σ x =sup{t:X t =x}. The main results of this paper are firstly that (i)c is slowly varying; (ii) $${{c\left( {X_t } \right)} \mathord{\left/ {\vphantom {{c\left( {X_t } \right)} {t\xrightarrow{\mathbb{P}}}}} \right. \kern-\nulldelimiterspace} {t\xrightarrow{\mathbb{P}}}}1;(iii) {{H_x } \mathord{\left/ {\vphantom {{H_x } {c(x)\xrightarrow{\mathbb{P}}}}} \right. \kern-\nulldelimiterspace} {c(x)\xrightarrow{\mathbb{P}}}}1;(iv) {{\sigma _x } \mathord{\left/ {\vphantom {{\sigma _x } {c(x)\xrightarrow{\mathbb{P}}1}}} \right. \kern-\nulldelimiterspace} {c(x)\xrightarrow{\mathbb{P}}1}}$$ are all equivalent: and secondly that (v) $${{c(X_t )} \mathord{\left/ {\vphantom {{c(X_t )} {t\xrightarrow{{a.s.}}}}} \right. \kern-\nulldelimiterspace} {t\xrightarrow{{a.s.}}}}1; (vi) {{H_x } \mathord{\left/ {\vphantom {{H_x } {c(x)\xrightarrow{{a.s.}}}}} \right. \kern-\nulldelimiterspace} {c(x)\xrightarrow{{a.s.}}}}1; (vii) {{\sigma _x } \mathord{\left/ {\vphantom {{\sigma _x } {c(x)\xrightarrow{{a.s.}}1}}} \right. \kern-\nulldelimiterspace} {c(x)\xrightarrow{{a.s.}}1}}$$ are all equivalent, and are implied by the condition $$\int\limits_{0 + } {c(x)^{ - 2} dv(x)〈 \infty } $$ . Other partial results for more general limit theorems are proved, and new results on regular variation are established.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01225825
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