Analogs of the arcsine distribution for sequences linearly generated by independent random variables

Yu. A. Davydov

Zapiski Nauchnykh Seminarov POMI, Problems of the theory of probability distributions. Part IV, Tome 72 (1977), pp. 62-74

Résumé

Let $\{\xi_k\}$, $k=\dots,-1,0,1,\dots$, be a sequence of independent identically distributed random variables with $E_{\xi_k}=0$, $D_{\xi_k}=\sigma^2<\infty$. Let $\{c_k\}$ be a numerical sequence such that $\sum^\infty_{-\infty}c^2_k<\infty$ Let $$ X_n=\sum^\infty_{-\infty}c_{k-n}\xi_k,\quad S_n=\sum^n_1X_k. $$ This article investigates the limit behavior of the distributions of functionals of the following type: $$ \nu_k=\dfrac1n\sum^n_1h(S_k), $$ where $h$ is a bounded function on $R^1$.