246 research outputs found
The Shape of Unlabeled Rooted Random Trees
We consider the number of nodes in the levels of unlabelled rooted random
trees and show that the stochastic process given by the properly scaled level
sizes weakly converges to the local time of a standard Brownian excursion.
Furthermore we compute the average and the distribution of the height of such
trees. These results extend existing results for conditioned Galton-Watson
trees and forests to the case of unlabelled rooted trees and show that they
behave in this respect essentially like a conditioned Galton-Watson process.Comment: 34 pages, 1 figur
Applications of uniformly distributed functions and sequences in statistic ergodic measuring techniques
AbstractThe stochastic ergodic measuring technique is used to measure voltages and mean values by a stochastic principle. After describing this principle it is shown how to estimate the measuring error deterministically with the discrepancy of functions and sequences
Degree distribution in random planar graphs
We prove that for each , the probability that a root vertex in a
random planar graph has degree tends to a computable constant , so
that the expected number of vertices of degree is asymptotically ,
and moreover that .
The proof uses the tools developed by Gimenez and Noy in their solution to
the problem of the asymptotic enumeration of planar graphs, and is based on a
detailed analysis of the generating functions involved in counting planar
graphs. However, in order to keep track of the degree of the root, new
technical difficulties arise. We obtain explicit, although quite involved
expressions, for the coefficients in the singular expansions of the generating
functions of interest, which allow us to use transfer theorems in order to get
an explicit expression for the probability generating function . From this we can compute the to any degree of accuracy, and derive
the asymptotic estimate for large values of ,
where is a constant defined analytically
- …