On the height of a random set of points in a -dimensional unit cube.
On the joint entropy of -wise-independent variables
How low can the joint entropy of -wise independent (for ) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than , for )? This question has been posed and partially answered in a recent work of Babai [Entropy versus pairwise independence (preliminary version),...
On the local time density of the reflecting Brownian bridge.
On the lower bound of the spectral norm of symmetric random matrices with independent entries.
On the Newcomb-Benford law in models of statistical data.
We consider positive real valued random data X with the decadic representation X = Σi=∞∞Di 10i and the first significant digit D = D(X) in {1,2,...,9} of X defined by the condition D = Di ≥ 1, Di+1 = Di+2 = ... = 0. The data X are said to satisfy the Newcomb-Benford law if P{D=d} = log10(d+1 / d) for all d in {1,2,...,9}. This law holds for example for the data with log10X uniformly distributed on an interval (m,n) where m and n are integers. We show that if log10X has a distribution function...
On the number of word occurrences in a semi-Markov sequence of letters
Let a finite alphabet Ω. We consider a sequence of letters from Ω generated by a discrete time semi-Markov process We derive the probability of a word occurrence in the sequence. We also obtain results for the mean and variance of the number of overlapping occurrences of a word in a finite discrete time semi-Markov sequence of letters under certain conditions.
On the shuffling algorithm for domino tilings.
On the size of a maximal induced tree in a random graph
On useful schema in survival analysis after heart attack
Recent model of lifetime after a heart attack involves some integer coefficients. Our goal is to get these coefficients in simple way and transparent form. To this aim we construct a schema according to a rule which combines the ideas used in the Pascal triangle and the generalized Fibonacci and Lucas numbers
One-dimensional random walks, decreasing rearrangements and discrete Steiner symmetrization
Orthogonal polynomial ensembles in probability theory.
Poisson convergence of numbers of vertices of a given degree in random graphs
The asymptotic distributions of the number of vertices of a given degree in random graphs, where the probabilities of edges may not be the same, are given. Using the method of Poisson convergence, distributions in a general and particular cases (complete, almost regular and bipartite graphs) are obtained.
Positive correlation for increasing events with disjoint dependencies does not imply positive correlation for all increasing events.
Poučka binomiální v počtu pravděpodobnosti
Preservation of log-concavity on summation
We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of...
Příspěvek k počtu pravděpodobnosti
Probabilistic proofs of hook length formulas involving trees.
Probabilities as values of modular forms and continued fractions.
Probability that an element of a finite group has a square root
Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 - ⌊√|G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 - 1/√|G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.
Processes on unimodular random networks.