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The Almost Sure Number of Pairwise Sums for Certain Random Integer Subsets Considered by P. Erdös

Michael J. Klass
Lecture Notes-Monograph Series
Vol. 35, Game Theory, Optimal Stopping, Probability and Statistics (2000), pp. 179-189
Stable URL: http://www.jstor.org/stable/4356089
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Almost Sure Number of Pairwise Sums for Certain Random Integer Subsets Considered by P. Erdös
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Abstract

Fix any λ > 0 and let X1, X2,... be independent and identically distributed 0-1 valued random variables such that $P(X_{j}=1)=\text{min}\left\{\sqrt{\frac{2\lambda}{\pi}\frac{\text{ln}\ j}{j}},1\right\}$. Let $G_{n}=\sum_{j=1}^{\lfloor n/2\rfloor }X_{j}X_{n-j}$. Gn is the number of times two numbers from the random set $S\equiv \{j\colon X_{j}=1\}$ add to n. We evaluate the almost sure limits $\underset n\rightarrow \infty \to{\text{lim inf}}\frac{G_{n}}{EG_{n}}\equiv c_{1}(\lambda)$ and $c_{2}(\lambda)\equiv \underset n\rightarrow \infty \to{\text{lim sup}}\frac{G_{n}}{EG_{n}}$, showing that $0\leq c_{1}(\lambda)<1

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