The large deviation principle for certain series

Arcones, Miguel A.

ESAIM: Probability and Statistics, Tome 8 (2004), pp. 200-220.

We study the large deviation principle for stochastic processes of the form $\{{\sum }_{k=1}^{\infty }{x}_k\left(t\right){\xi }_k:t\in T\}$, where ${\left\{{\xi }_k\right\}}_{k=1}^{\infty }$ is a sequence of i.i.d.r.v.’s with mean zero and $x_k\left(t\right)\in ℝ$. We present necessary and sufficient conditions for the large deviation principle for these stochastic processes in several situations. Our approach is based in showing the large deviation principle of the finite dimensional distributions and an exponential asymptotic equicontinuity condition. In order to get the exponential asymptotic equicontinuity condition, we derive new concentration inequalities, which are of independent interest.

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