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The twin prime conjecture
This is an exposition of recent developments in the theory of bounded
differences between primes. Readers are expected to be beginners of analytic
number theory.
The present text is a substantially improved and augmented version of the one
that I had prepared for my talk which I delivered at the Annual Meeting of the
Mathematical Society of Japan on 15 March 2014.Comment: 17 page
On the paper ``Weak convergence of some classes of martingales with jumps''
This note extends some results of Nishiyama [Ann. Probab. 28 (2000)
685--712]. A maximal inequality for stochastic integrals with respect to
integer-valued random measures which may have infinitely many jumps on compact
time intervals is given. By using it, a tightness criterion is obtained; if the
so-called quadratic modulus is bounded in probability and if a certain entropy
condition on the parameter space is satisfied, then the tightness follows. Our
approach is based on the entropy techniques developed in the modern theory of
empirical processes.Comment: Published at http://dx.doi.org/10.1214/009117906000000755 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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