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.

scholarly journals Some results using general moment functions

1969 ◽  
Vol 10 (3-4) ◽  
pp. 429-441 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Random Variables ◽  
Closure Property ◽  
Probability Generating Functions ◽  
Moment Functions ◽  
Result Theorem ◽  
Independent And Identically Distributed

SummaryLet {Xn} be a sequence fo independent and identically distributed random variables such that 0 <μ = εXn ≦ + ∞ and write Sn = X1+X2+ … +Xn. Letv ≧ 0 be an integer and let M(x) be a non-decreasing function of x ≧ 0 such that M(x)/x is non-increasing and M(0) > 0. Then if ε|X1νM(|X1|) < ∞ and μ < ∞ it follows that ε|Sn|νM(|Sn|) ~ (nμ)vM(nμ) as n → ∞. If μ = ∞ (ν = 0) then εM(|Sn|) = 0(n). A variety of results stem from this main theorem (Theorem 2), concerning a closure property of probability generating functions and a random walk result (Theorem 1) connected with queues.

ON THE DISCRETE-TIME G/GI/∞ QUEUE

2008 ◽  
Vol 22 (4) ◽  
pp. 557-585 ◽  
Author(s):  
Iddo Eliazar
Keyword(s):  
Fixed Points ◽  
Discrete Time ◽  
Generating Functions ◽  
Random Variables ◽  
Output Process ◽  
Input Output ◽  
Input Process ◽  
Probability Generating Functions ◽  
Queue Model ◽  
Multidimensional Probability

The discrete-time G/GI/∞ queue model is explored. Jobs arrive to an infinite-server queuing system following an arbitrary input process X; job sizes are general independent and identically distributed random variables. The system's output process Y (of job departures) and queue process N (tracking the number of jobs present in the system) are analyzed. Various statistics of the stochastic maps X↦ Y and X↦ N are explicitly obtained, including means, variances, autocovariances, cross-covariances, and multidimensional probability generating functions. In the case of stationary inputs, we further compute the spectral densities of the stochastic maps, characterize the fixed points (in the L2 sense) of the input–output map, precisely determine when the output and queue processes display either short-ranged or long-ranged temporal dependencies, and prove a decomposition result regarding the intrinsic L2 structure of general stationary G/GI/∞ systems.

scholarly journals Probability generating functions for discrete real-valued random variables

10.4213/tvp8 ◽  
2007 ◽  
Vol 52 (1) ◽  
pp. 129-149
Author(s):  
Manuel Leote Tavares Ingles Esquivel ◽  
Manuel Leote Tavares Ingles Esquivel
Keyword(s):  
Generating Functions ◽  
Random Variables ◽  
Probability Generating Functions

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.

Sooner and later waiting time problems for patterns in Markov dependent trials

2003 ◽  
Vol 40 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Qing Han ◽  
Katuomi Hirano
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Random Variables ◽  
Probability Functions ◽  
Probability Generating Functions ◽  
Markov Dependent Trials

In this paper, we investigate sooner and later waiting time problems for patterns S0 and S1 in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences of S0 and between successive occurrences of S0 and S1 and of the waiting time until the rth occurrence of S0 are also given.

Branching and clustering models associated with the ‘lost-games distribution’

1969 ◽  
Vol 6 (3) ◽  
pp. 700-703 ◽  
Author(s):  
Adrienne W. Kemp ◽  
C. D. Kemp
Keyword(s):  
Generating Functions ◽  
Random Variables ◽  
Probability Generating Functions ◽  
Generalized Distributions

We use Gurland's (1957) definition and notation for generalized distributions, i.e., given random variables Xi with probability generating functions gi(s), i = 1, 2, 3, if g3(s) = g1[g2(s)], we say that X3 is X1 generalized by X2 and write X3~ X1VX2.

Sooner and later waiting time problems for patterns in Markov dependent trials

2003 ◽  
Vol 40 (01) ◽  
pp. 73-86 ◽  
Author(s):  
Qing Han ◽  
Katuomi Hirano
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Random Variables ◽  
Probability Functions ◽  
Probability Generating Functions ◽  
Markov Dependent Trials

In this paper, we investigate sooner and later waiting time problems for patterns S 0 and S 1 in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences of S 0 and between successive occurrences of S 0 and S 1 and of the waiting time until the rth occurrence of S 0 are also given.

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