The distributions of some characteristics of random walks and related combinatorial identities
نویسندگان
چکیده
In this talk, we show an interesting fact that a quarter of paths of random walks of any length n(≥ 2) have two maximums. Moreover, it holds that the asymptotic distribution of number of maximums of paths obeys the geometric distribution of parameter 1/2. Furthermore, we provide the generating function for counting number of paths jointly with number k of maximums and number l of minimums. This result comes from the careful consideration of paths with restricted width by using the combinatorial symbolic method. We can extend to other topics on number of zero returns, zero touches, etc. for paths of random walks. As by-product, we obtain many combinatorial identities. 1 Distribution of number of maximums in random walks Dyck paths are characterized as sequences of numbers x = (x0, x1, . . . , x2n) satisfying the conditions: x0 = x2n = 0, xj ≥ 0, |xj+1 − xj | = 1 for 0 ≤ j ≤ 2n− 1 . (1.1) It is common that Dyck paths of length 2n correspond one-to-one to binary trees having n internal nodes, and can be enumerated by Catalan numbers. Let D be the class of Dyck paths, the size of which is defined by length. Then, we can symbolically express the class as follows: D = φ+ (↗ D ↘)×D, (1.2) or, D = φ+ (↗ D ↘) + (↗ D ↘) + . . . , (1.3) where ↗ denotes an ascent step, and ↘ denotes a descent step. ∗The University of Electro-Communications, Tokyo, JAPAN, E-mail: [email protected] †The University of Electro-Communications,Tokyo,JAPAN,E-mail:[email protected] ‡The University of Electro-Communications, Tokyo, JAPAN, E-mail: [email protected]
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تاریخ انتشار 2015