Random fractals and tree-indexed Markov chains

Random Fractals and Tree - Indexed Markov

We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrot's...

Some Limit Properties of Random Transition Probability for Second-Order Nonhomogeneous Markov Chains Indexed by a Tree

Random walks and Markov chains

This lecture discusses Markov chains, which capture and formalize the idea of a memoryless random walk on a finite number of states, and which have wide applicability as a statistical model of many phenomena. Markov chains are postulated to have a set of possible states, and to transition randomly from one state to a next state, where the probability of transitioning to a particular next state ...

Markov Chains and Random Walks

Let G = (V,E) be a connected, undirected graph with n vertices and m edges. For a vertex v ∈ V , Γ(v) denotes the set of neighbors of v in G. A random walk on G is the following process, which occurs in a sequence of discrete steps: starting at a vertex v0, we proceed at the first step to a random edge incident on v0 and walking along it to a vertex v1, and so on. ”Random chosen neighbor” will ...

Tree-Structured Like Representations for Continuous and Discrete Indexed Markov Random Fields

Markov random fields (MRFs) are ubiquitous models for conveniently capturing complex statistical dependencies given a collection of random variables. For example, when modeling the temperature distribution of a continuously indexed region, it is common to assume that the temperature at one point depends on some differential neighborhood. MRFs can capture such dependencies using a set of conditi...