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
2005 ◽  
Vol DMTCS Proceedings vol. AD,... (Proceedings) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher

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.


2009 ◽  
Vol Vol. 11 no. 1 (Combinatorics) ◽  
Author(s):  
Charlotte Brennan ◽  
Arnold Knopfmacher

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.


2013 ◽  
Vol 20 (5) ◽  
pp. 415-449 ◽  
Author(s):  
S. T. Tse ◽  
P. A. Forsyth ◽  
J. S. Kennedy ◽  
H. Windcliff

2013 ◽  
Vol 110 (3) ◽  
pp. 621-639 ◽  
Author(s):  
Bryan M. Krause ◽  
Matthew I. Banks

The neural mechanisms of sensory responses recorded from the scalp or cortical surface remain controversial. Evoked vs. induced response components (i.e., changes in mean vs. variance) are associated with bottom-up vs. top-down processing, but trial-by-trial response variability can confound this interpretation. Phase reset of ongoing oscillations has also been postulated to contribute to sensory responses. In this article, we present evidence that responses under passive listening conditions are dominated by variable evoked response components. We measured the mean, variance, and phase of complex time-frequency coefficients of epidurally recorded responses to acoustic stimuli in rats. During the stimulus, changes in mean, variance, and phase tended to co-occur. After the stimulus, there was a small, low-frequency offset response in the mean and modest, prolonged desynchronization in the alpha band. Simulations showed that trial-by-trial variability in the mean can account for most of the variance and phase changes observed during the stimulus. This variability was state dependent, with smallest variability during periods of greatest arousal. Our data suggest that cortical responses to auditory stimuli reflect variable inputs to the cortical network. These analyses suggest that caution should be exercised when interpreting variance and phase changes in terms of top-down cortical processing.


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