scholarly journals The distribution of ascents of size $d$ or more in samples of geometric random variables

2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Nonnegative Integer ◽  
Random Variables ◽  
Limiting Distribution ◽  
Geometric Probability ◽  
International Audience ◽  
The Mean ◽  
Mean Variance

International audience 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.

scholarly journals The distribution of ascents of size d or more in compositions

2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Positive Integer ◽  
Nonnegative Integer ◽  
Finite Sequence ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean ◽  
Average Size ◽  
Positive Integers ◽  
Mean Variance

Combinatorics International audience A composition of a positive integer n is a finite sequence of positive integers a(1), a(2), ..., a(k) such that a(1) + a(2) + ... + a(k) = n. Let d be a fixed nonnegative integer. We say that we have an ascent of size d or more if a(i+1) >= a(i) + d. We determine the mean, variance and limiting distribution of the number of ascents of size d or more in the set of compositions of n. We also study the average size of the greatest ascent over all compositions of n.

scholarly journals The Limiting Distribution for the Number of Symbol Comparisons Used by QuickSort is Nondegenerate (Extended Abstract)

2012 ◽  
Vol DMTCS Proceedings vol. AQ,... (Proceedings) ◽  
Author(s):  
Patrick Bindjeme ◽  
james Allen fill
Keyword(s):  
Large Class ◽  
Continuous Time ◽  
Random Variable ◽  
Limiting Distribution ◽  
International Audience ◽  
The Mean

International audience In a continuous-time setting, Fill (2012) proved, for a large class of probabilistic sources, that the number of symbol comparisons used by $\texttt{QuickSort}$, when centered by subtracting the mean and scaled by dividing by time, has a limiting distribution, but proved little about that limiting random variable $Y$—not even that it is nondegenerate. We establish the nondegeneracy of $Y$. The proof is perhaps surprisingly difficult.

scholarly journals The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance

2006 ◽  
Vol 16 (4) ◽  
pp. 2195-2214 ◽  
Author(s):  
Michael G. B. Blum ◽  
Olivier François ◽  
Svante Janson
Keyword(s):  
Phylogenetic Tree ◽  
Limiting Distribution ◽  
The Mean ◽  
Mean Variance

scholarly journals Some remarks on the mean value of the Riemann zeta-function and other Dirichlet series 1

1978 ◽  
Vol Volume 1 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Zeta Function ◽  
Dirichlet Series ◽  
Positive Constant ◽  
Riemann Zeta Function ◽  
Nonnegative Integer ◽  
Mean Value ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta ◽  
The Mean

International audience The present paper is concerned with $\Omega$-estimates of the quantity $$(1/H)\int_{T}^{T+H}\vert(d^m/ds^m)\zeta^k(\frac{1}{2}+it)\vert dt$$ where $k$ is a positive number (not necessarily an integer), $m$ a nonnegative integer, and $(\log T)^{\delta}\leq H \leq T$, where $\delta$ is a small positive constant. The main theorems are stated for Dirichlet series satisfying certain conditions and the corollaries concerning the zeta function illustrate quite well the scope and interest of the results. %It is proved that if $2k\geq1$ and $T\geq T_0(\delta)$, then $$(1/H)\int_{T}^{T+H}\vert \zeta(\frac{1}{2}+it)\vert^{2k}dt > (\log H)^{k^2}(\log\log H)^{-C}$$ and $$(1/H)\int_{T}^{T+H} \vert\zeta'(\frac{1}{2}+it)\vert dt > (\log H)^{5/4}(\log\log H)^{-C},$$ where $C$ is a constant depending only on $\delta$.

scholarly journals Distribution of the Number of Encryptions in Revocation Schemes for Stateless Receivers

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Christopher Eagle ◽  
Zhicheng Gao ◽  
Mohamed Omar ◽  
Daniel Panario ◽  
Bruce Richmond
Keyword(s):  
Difference Scheme ◽  
Limiting Distribution ◽  
Broadcast Encryption ◽  
Discrete Algorithms ◽  
Encryption Schemes ◽  
International Audience ◽  
The Mean ◽  
Subset Difference

International audience We study the number of encryptions necessary to revoke a set of users in the complete subtree scheme (CST) and the subset-difference scheme (SD). These are well-known tree based broadcast encryption schemes. Park and Blake in: Journal of Discrete Algorithms, vol. 4, 2006, pp. 215―238, give the mean number of encryptions for these schemes. We continue their analysis and show that the limiting distribution of the number of encryptions for these schemes is normal. This implies that the mean numbers of Park and Blake are good estimates for the number of necessary encryptions used by these schemes.

scholarly journals Descent variation of samples of geometric random variables

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher
Keyword(s):  
Generating Functions ◽  
Random Variables ◽  
Geometric Probability ◽  
Probability Generating Functions ◽  
International Audience

Combinatorics International audience In this paper, we consider random words ω1ω2ω3⋯ωn of length n, where the letters ωi ∈ℕ are independently generated with a geometric probability such that Pωi=k=pqk-1 where p+q=1 . We have a descent at position i whenever ωi+1 < ωi. The size of such a descent is ωi-ωi+1 and the descent variation is the sum of all the descent sizes for that word. We study various types of random words over the infinite alphabet ℕ, where the letters have geometric probabilities, and find the probability generating functions for descent variation of such words.

$\mathcal{L}^p$-PROJECTIONS OF RANDOM VARIABLES AND ITS APPLICATION TO FINANCE

2008 ◽  
Vol 11 (08) ◽  
pp. 869-888 ◽  
Author(s):  
TAKUJI ARAI
Keyword(s):  
Random Variables ◽  
Contingent Claim ◽  
Text Setting ◽  
Hedging Strategy ◽  
Unique Existence ◽  
Optimal Hedging ◽  
Suitable Space ◽  
The Mean ◽  
Hedging Problem ◽  
Mean Variance

The aim of this paper is to give an extension of the mean-variance hedging problem to the [Formula: see text]-setting, where 1 < p < ∞. Remark that the mean-variance hedging is corresponding to the case where p = 2. Firstly, we prove that the unique existence of the optimal hedging strategy in the [Formula: see text]-sense, which is the [Formula: see text]-projection of the underlying contingent claim onto a suitable space of stochastic integrations. Next, we obtain its feedback representation under some additional assumptions. Moreover, the valuation problem induced by the [Formula: see text]-projections naturally is discussed.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

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