An Introduction to Scientific Communication by Перекальская Т.К., Выгонская Л.Н. (ред.)

By Перекальская Т.К., Выгонская Л.Н. (ред.)

Учебное пособие ставит своей задачей формирование навыков научной коммуникации на английском языке с целью оптимизации лингвистического сообщения и предназначено для студентов и аспирантов механико-математического факультета МГУ.

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4 The complementary distribution functions are regularly varying, and the on- and off-times have infinite variance, cf. [130, 216]: 1. F on (x) = x −αon Lon (x), 2. F off (x) = x −αoff Loff (x), 3. F on (x) = o(F off (x)) as x → ∞ with αon ∈ (1, 2); with αoff ∈ (1, 2); or F off (x) = o(F on (x)) as x → ∞, where Lon (·) and Loff (·) are slowly varying at ∞. In the sequel, we use the following identification: H ≡ 3 − min(αon , αoff ) . 2 Above we already considered the limit n → ∞; now we let, in addition, also T grow large.

1 (i) Consider the path f (·) that is equal to 0 for t < 0, equal to κt for t ∈ [0, T ], and equal to κT for t > T . What is the ‘Brownian norm’ of this path? 2. Schilder’s theorem S := St = {f : ∃t ≥ 0 : f (t) ≥ a + νt}. t≥0 Assume that S is most likely reached through a straight line from the origin (which is reasonable, as the increments are independent). Find the most likely path. Solution. (i) f (t) = κ for t ∈ (0, T ) and 0 otherwise. Hence T I(f ) = 0 1 2 1 κ dt = κ 2 T . 2 2 (ii) A straight line with slope κ reaches a + νt after a/(κ − ν) time (notice that this requires that κ is larger than ν).

Also the convergence to α-stable L´evy motion, with the limits taken in reverse order, is established in that paper; see also [288]. A recent article by Mikosch, Resnick, Rootz´en, and Stegeman [216] explores how these two limits relate; in fact it is proven that under slow growth (connection rates are modest relative to the connection length distribution tails) the limit is an α-stable L´evy motion, whereas under fast growth (the opposite situation) there is an fBm limit. In this respect, also the recent work (on the so-called ‘telecom process’) by Kaj and Taqqu [144] should be mentioned.

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