The Maximum of a Random Walk and Its Application to Rectangle Packing
نویسندگان
چکیده
Let S0; : : : ; Sn be a symmetric random walk that starts at the origin (S0 = 0), and takes steps uniformly distributed on [ 1;+1]. We study the large-n behavior of the expected maximum excursion and prove the estimate E max 0 k n Sk =r2n 3 c+ 1 5r 2 3 n 1=2 +O(n 3=2); where c = 0:297952 : : :. This estimate applies to the problem of packing n rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height, n4 + 12Emax0 j n Sj + 1 2 = n4 + O(n1=2); when the rectangle sides are 2n independent uniform random draws from [0; 1].
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