A new maximal inequality and invariance principle for stationary sequences
Let \((X_i)_{i\in\mathbb Z}\) be a stationary sequence of centered random variables with finite second moment. Denote by \({\mathcal F}_k\) the \(\sigma\)-field generated by \(\{X_i,i\leq k\}\), and define \(S_n=\sum^n_{i=1}X_i\), \(W_n(t)=S_{[nt]}/\sqrt n\), \(0\leq t\leq 1\), where \([x]\) denotes the integer part of \(x\). Finally, let \(\{W(t),0\leq t\leq 1\}\) be a standard Brownian motion. Assume that \(\sum^\infty_{n=1}\| E(S_n\mid {\mathcal F}_0)\|/n^{3/2}<\infty\), where \(\| X\|=\sqrt {E(X^2)}\). Then, \(W_n(t)\) converges weakly to \(\sqrt\eta W(t)\), as \(n\to\infty\), where \(\eta\) is a nonnegative random variable with finite mean \(E[\eta]=\sigma^2\) and independent of \(\{W(t),0\leq t\leq 1\}\). Moreover, the variable \(\eta\) is such that \(\lim_{n\to\infty} E(S^2_n\mid \Omega)/n=\eta\) in \(L_1\) where \(\Omega\) is the invariant sigma field. In particular, \(\lim_{n\to\infty} E(S^2_n)n=\sigma^2\).
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