کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
7550176 | 1489923 | 2018 | 8 صفحه PDF | دانلود رایگان |
عنوان انگلیسی مقاله ISI
A sharp bound on the expected number of upcrossings of an L2-bounded Martingale
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کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه
ریاضیات
ریاضیات (عمومی)
پیش نمایش صفحه اول مقاله
چکیده انگلیسی
For a martingale M starting at x with final variance Ï2, and an interval (a,b), let Î=bâaÏ be the normalized length of the interval and let δ=|xâa|Ï be the normalized distance from the initial point to the lower endpoint of the interval. The expected number of upcrossings of (a,b) by M is at most 1+δ2âδ2Î if Î2â¤1+δ2 and at most 11+(Î+δ)2 otherwise. Both bounds are sharp, attained by Standard Brownian Motion stopped at appropriate stopping times. Both bounds also attain the Doob upper bound on the expected number of upcrossings of (a,b) for submartingales with the corresponding final distribution. Each of these two bounds is at most Ï2(bâa), with equality in the first bound for δ=0. The upper bound Ï2 on the length covered by M during upcrossings of an interval restricts the possible variability of a martingale in terms of its final variance. This is in the same spirit as the Dubins & Schwarz sharp upper bound Ï on the expected maximum of M above x, the Dubins & Schwarz sharp upper bound Ï2 on the expected maximal distance of M from x, and the Dubins, Gilat & Meilijson sharp upper bound Ï3 on the expected diameter of M.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Stochastic Processes and their Applications - Volume 128, Issue 6, June 2018, Pages 1849-1856
Journal: Stochastic Processes and their Applications - Volume 128, Issue 6, June 2018, Pages 1849-1856
نویسندگان
David Gilat, Isaac Meilijson, Laura Sacerdote,