Ultimate luminosities and energies of photon colliders

A photon collider luminosity and its energy are determined by the parameters of an electron-electron linear collider (energy, power, beam emittances) and collision effects. The main collision effect is the coherent e+e- pair creation. At low energies (2E < 0.5 - 1 TeV) this process is suppressed due to repulsion of electron beams. In this region gamma-gamma luminosity above 10^{35} cm-2 s-1 is possible (10^{33} is sufficient). At higher energies the limited average beam power and coherent pair creation restrict the maximum energy of photon colliders (with sufficient luminosity) at 2E ~ 5 TeV. Obtaining high luminosities requires the development of new methods of production beams with low emittances such as a laser cooling.Comment: 15 pages, Latex, 6 figures(eps), submitted to Proceedings of the Symposium on Future High Energy Colliders, ITP, UCSB, Santa Barbara, October 1996. AIP Pres

Marcinkiewicz-Zygmund Strong Law of Large Numbers for Pairwise i.i.d. Random Variables

It is shown that the Marcinkiewicz-Zygmund strong law of large numbers holds for pairwise independent identically distributed random variables. It is proved that if $X_{1}, X_{2}, \ldots$ are pairwise independent identically distributed random variables such that $E|X_{1}|^p < \infty$ for some $1 < p < 2$, then $(S_{n}-ES_{n})/n^{1/p} \to 0$ a.s. where $S_{n} = \sum_{k=1}^{n} X_{k}$