  The violation of Bell inequality not only provides the most radical departure
of quantum theory from classical concepts, but also paves the way of
applications in such as device independent randomness certification. Here, we
derive the tight upper bound of the maximum quantum value for chained Bell
inequality with arbitrary number of measurements on each party. \lxh{ The
constraints where the upper bound saturates are also presented. This method
provides us the necessary and sufficient conditions for some quantum states to
violate the chained Bell inequality with arbitrary number of measurements}.
Based on the tight upper bound we present the lower bounds on the device
independent randomness with respect to the Werner states. \lxh{In particular,
we present lower bounds on the randomness generation rates of chained Bell
inequality for different number of measurements, which are compared with the
family of Bell inequalities proposed by Wooltorton et al. [Phys. Rev. Lett.
129, 150403 (2022)]. Our results show that chained Bell inequality with three
measurements has certain advantages at a low level of noise and could be used
to improve randomness generation rates in practice.
