Linear delay differential equations, Asymptotic behavior, Sums of independent random variables, Random walks
Let t⩾0. Select numbers randomly from the interval [0,1] until the sum is greater than t . Let α(t) be the expected number of selections. We prove that α(t)=et for 0⩽t⩽1. Moreover, . This limit is a special case of our asymptotic results for solutions of the delay differential equation f′(t)=f(t)-f(t-1) for t>1. We also consider four other solutions of this equation that are related to the above selection process.
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This is the authors' post print version of the paper, the published version can be found at the links below.
Ćurgus, Branko and Jewett, Robert I., "An Unexpected Limit of Expected Values" (2007). Mathematics. 70.