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An approximation of partial sums of independent RV'-s, and the sample DF. I
SummaryLet Sn=X1+X2+⋯+Xnbe the sum of i.i.d.r.v.-s, EX1=0, EX12=1, and let Tn= Y1+Y2+⋯+Ynbe the sum of independent standard normal variables. Strassen proved in [14] that if X1 has a finite fourth
Blow-up Lemma
Regular pairs behave like complete bipartite graphs from the point of view of bounded degree subgraphs.
The eigenvalues of random symmetric matrices
TLDR
It is shown that with probability 1-o(1)all eigenvalues belong to the above intervalI if μ=0, while in case μ>0 only the largest eigenvalueλ1 is outsideI, and λ1 asymptotically has a normal distribution with expectation (n−1)μ+v+(σ2/μ) and variance 2σ2 (bounded variance!).
Szemeredi''s Regularity Lemma and its applications in graph theory
Szemer\''edi''s Regularity Lemma is an important tool in discrete mathematics. It says that, in some sense, all graphs can be approximated by random-looking graphs. Therefore the lemma helps in
A Note on Ramsey Numbers
A generalization of a problem of Steinhaus
This is the fixed version of an article made available by an organization that acts as a publisher by formally and exclusively declaring the article "published". If it is an "early release" article
On the probability that a random ±1-matrix is singular
We report some progress on the old problem of estimating the probability, Pn, that a random n× n ± 1 matrix is singular: Theorem. There is a positive constant ε for which Pn < (1− ε)n. This is a
Storing a sparse table with O(1) worst case access time
TLDR
A data structure for representing a set of n items from a universe of m items, which uses space n+o(n) and accommodates membership queries in constant time and is easy to implement.
On optimal matchings
TLDR
Givenn random red points on the unit square, the transportation cost between them is tipically √n logn, where logn is the number of red points in the ellipsoidal plane.
Shrinking in a Growing Economy? The Mystery of Physical Stature during the Industrial Revolution
Reviews the evidence on early-industrial height cycles and shows why the economic transition put downward pressure on the nutritional status of the European and American populations.(This abstract
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