Density estimation for linear processes

Summary

LetX_t, ...,X_n be random variables forming a realization from a linear process$X_t = \sum\limits_{r = 0}^\infty {g_r Z_{t - r} }$ where {Z_t} is a sequence of independent and identically distributed random variables with E|Z_t|<∞ for some ε>0, andg_r→0 asr→∞ at some specified rate. LetX₁ have a probability density functionf. It is then established that for every realx, the standard kernel type estimator$\hat f_n (x)$ based onX_t (1≦t≦n) is, under some general regularity conditions, asymptotically normal and converges a.s. tof(x) asn→∞.