Let n ∈ N and X(n) = (X 1 (n), . . . , X d(n) (n)) be a sequence of random vectors. We prove that, under certain dependence conditions, the cdf of the maximum of X i (n) asymptotically equals to the cdf of the maximum of a random vector with the same but independent marginal distributions. To prove our result on extremal independence, we obtain new lower and upper bounds on the probability that none of a given finite set of events occurs. Using our result, we show that, under certain conditions, including Berman-type condition, a sequence of Gaussian random vectors possesses the extremal independence property. We also prove that certain extremal characterstics of binomial random graphs and hypergraphs, after an appropriate rescaling, have asymptotical Gumbel distribution (in particular, maximum codegrees in random hypergraphs and the maximum number of cliques sharing a given vertex in random graphs).

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Traditional statistical learning theory relies on the assumption that data are identically and independently generated from a given distribution (i.i.d.). The independently distributed assumption, on the other hand, fails to hold in many real applications. In this survey, we consider learning settings in which examples are dependent and their dependence relationship can be characterised by a graph. We collect various graph-dependent concentration bounds, which are then used to derive Rademacher and stability generalization bounds for learning from graph-dependent data. We illustrate this paradigm with three learning tasks and provide some research directions for future work. To the best of our knowledge this is the first survey on this subject.

Let integer n3 and integer r = r(n) 3. Define the binomial random r-uniform hypergraph H r (n, p) to be the r-uniform graph on the vertex set [n] such that each rset is an edge independently with probability p. A hypergraph is linear if every pair of hyperedges intersects in at most one vertex. We study the probability of linearity of random hypergraphs H r (n, p) via cluster expansion and give more precise asymptotics of the probability in question, improving the asymptotic probability of linearity obtained by McKay and Tian, in particular, when r = 3 and p = o(n −7/5 ).