Geodesic
The distribution of ascents of size $d$ or more in samples of geometric random variables
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AD, International Conference on Analysis of Algorithms, DMTCS Proceedings vol. AD, International Conference on Analysis of Algorithms (2005).

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We consider words or strings of characters $a_1a_2a_3 \ldots a_n$ of length $n$, where the letters $a_i \in \mathbb{Z}$ are independently generated with a geometric probability $\mathbb{P} \{ X=k \} = pq^{k-1}$ where $p+q=1$. Let $d$ be a fixed nonnegative integer. We say that we have an ascent of size $d$ or more if $a_{i+1} \geq a_i+d$. We determine the mean, variance and limiting distribution of the number of ascents of size $d$ or more in a random geometrically distributed word. 
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     author = {Brennan, Charlotte and Knopfmacher, Arnold},
     title = {The distribution of ascents of size $d$ or more in samples of geometric random variables},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AD, International Conference on Analysis of Algorithms},
     year = {2005},
     doi = {10.46298/dmtcs.3382},
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     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3382/}
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Brennan, Charlotte; Knopfmacher, Arnold. The distribution of ascents of size $d$ or more in samples of geometric random variables. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AD, International Conference on Analysis of Algorithms, DMTCS Proceedings vol. AD, International Conference on Analysis of Algorithms (2005). doi : 10.46298/dmtcs.3382. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3382/

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