RANDOM WALKS AT RANDOM TIMES: CONVERGENCE TO ITERATED LEVY MOTION, FRACTIONAL STABLE MOTIONS, AND OTHER SELF-SIMILAR PROCESSES

Cited 2 time in webofscience Cited 0 time in scopus
  • Hit : 406
  • Download : 0
For a random walk defined for a doubly infinite sequence of times, we let the time parameter itself be an integer-valued process, and call the orginal process a random walk at random time. We find the scaling limit which generalizes the so-called iterated Brownian motion. Khoshnevisan and Lewis [Ann. Appl. Probab. 9 (1999) 629-667] suggested "the existence of a form of measure-theoretic duality" between iterated Brownian motion and a Brownian motion in random scenery. We show that a random walk at random time can be considered a random walk in "alternating" scenery, thus hinting at a mechanism behind this duality. Following Cohen and Samorodnitsky [Ann. Appl. Probab. 16 (2006) 1432-1461], we also consider alternating random reward schema associated to random walks at random times. Whereas random reward schema scale to local time fractional stable motions, we show that the alternating random reward schema scale to indicator fractional stable motions. Finally, we show that one may recursively "subordinate" random time processes to get new local time and indicator fractional stable motions and new stable processes in random scenery or at random times. When alpha = 2, the fractional stable motions given by the recursion are fractional Brownian motions with dyadic H is an element of (0, 1). Also, we see that "un-subordinating" via a time-change allows one to, in some sense, extract Brownian motion from fractional Brownian motions with H < 1/2
Publisher
INST MATHEMATICAL STATISTICS
Issue Date
2013-07
Language
English
Article Type
Article
Citation

ANNALS OF PROBABILITY, v.41, no.4, pp.2682 - 2708

ISSN
0091-1798
DOI
10.1214/12-AOP770
URI
http://hdl.handle.net/10203/213194
Appears in Collection
MA-Journal Papers(저널논문)
Files in This Item
There are no files associated with this item.
This item is cited by other documents in WoS
⊙ Detail Information in WoSⓡ Click to see webofscience_button
⊙ Cited 2 items in WoS Click to see citing articles in records_button

qr_code

  • mendeley

    citeulike


rss_1.0 rss_2.0 atom_1.0