We study the statistical behavior of the 1D nonlinear \beta-FPU chain.

When the nonlinearity is weak the thermal equilibrium state of the

system can be described as weakly interacting linear waves, and

the wave action n_k is proportional to 1/w_k with w_k - linear

dispersion (Rayleigh-Jeans distribution).We demonstrate numerically that

surprisingly even in the strongly nonlinear limit the system still

can be effectively described like weakly interacting waves and hence

the same distribution holds. This arises because strong nonlinearity

effectively renormalizes the linear dispersion frequency. We find

the theoretical prediction for the frequency renormalizing factor and

show that it is in good agreement with numerical experiments. Moreover

the presence of nonlinearity widens the resonance peaks which results

in the occurrence of near-resonance wave-wave interactions due to

discreteness.

We also employ Random Phase Approximation

and use the Wiener-Khinchin formula and the perturbation methods to

derive the form of the resonance width as a function of the wave

number and the nonlinearity strength. The resulting analytical

prediction for the resonance width is in good agreement with

numerical experiments.Finally we numerically observed that in the

thermodynamical equilibrium \beta-FPU exhibits spatially localized

oscillations - Discrete Breathers.