Let E ⊆ F and E′ ⊆ F ′ be Borel equivalence relations on the standard Borel spaces X and Y , respectively. The pair (E,F ) is simultaneously Borel reducible to the pair (E′, F ′) if there is a Borel function f : X → Y that is both a reduction from E to E′ and a reduction from F to F ′. Simultaneous Borel embeddings and isomorphisms are defined analogously. We classify all pairs E ⊆ F of smooth ...

In this note we will study upper bounds of random variables of the type sup A∈A |ν n (A) − ν(A)| , where A is a class of sets that needs to fulll certain assumptions. These bounds are important tools in the analysis of learning processes and probabilistic theories of pattern recognition. The presentation given here is based on [DGL96].

Let A(t) denote the cluster produced by internal diffusion limited aggregation (internal DLA) with t particles in dimension d ≥ 3. We show that A(t) is approximately spherical, up to an O( √ log t) error. In the process known as internal diffusion limited aggregation (internal DLA) one constructs for each integer time t ≥ 0 an occupied set A(t) ⊂ Zd as follows: begin with A(0) = ∅ and A(1) = {0...

This thesis establishes some results in algorithmic information theory. Martin-Löf in 1966 formalized the notion of what it means for a given sequence to be random, using the theory of computation. Equivalent, alternate definitions justify that this definition is robust. A finite string is called random if there is no short program which outputs the string that is, the Kolmogorov complexity of ...

These notes are based on the two lectures1 I gave in Stat 501, Probability Theory I, as a substitute for Professor Cheng Ouyang. There is a brief discussion of the so-called “basic grouping lemma” concerning independent σ-algebras, followed by a more detailed discussion of the Borel–Cantelli lemma, some applications, and some elaborations on independence. Independence, and the basic grouping le...

Oracle separation methods are used in cryptography to rule out blackbox reductions between cryptographic primitives. It is sufficient to find an oracle relative to which the base primitive exists but there are no secure instances of the constructed primitive. In practice, it is beyond our current reach to construct a fixed oracle with such properties for most of the reductions because it is dif...

Let S∞ be the group of all permutations of N and X be a standard Borel space. Then the space XN of functions from N to X is a standard Borel space, and S∞ acts on this space by permutation where given x ∈ XN and g ∈ S∞, g · x(n) = x(g−1(n)). Given any countable subgroup G of S∞, we can likewise restrict this action to G and consider the induced orbit equivalence relation on XN, which we will no...