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Some applications of the Strassen law of iterated logarithm for normalized integrals of processes with weak dependence. (English. Ukrainian original) Zbl 0960.60030
Theory Probab. Math. Stat. 59, 11-19 (1999); translation from Teor. Jmorvirn. Mat. Stat. 59, 10-19 (1998).
For a strictly stationary random process \(X(t)\) with \(E X(t)=0\), \(E {}X(t){}^5<\infty\) which satisfies the \(\varphi\)-mixing condition with \( \int_0^\infty \varphi^{1/2}(\tau) d\tau<\infty\), Strassen’s law of iterated logarithm is proved. I.e., it is shown that for a sequence of processes \[ \zeta_n(t)={1\over \sqrt{2\sigma^2 n\ln\ln n}}\int_0^{nt}X(s) ds, \] where \(\sigma^2=2\int_0^{\infty} E X(0)X(t) dt\), the set of limit points in \(C[0,1]\) as \(n\to\infty\) is the set of absolutely continuous functions \(x(t)\) such that \(x(0)=0\), \(\int_0^1(x'(t))^2 dt\leq 1\). This result is applied to analyze the asymptotic behavior of solutions of the differential equations \(Z_{n}'(t)={1\over n}[a(t,Z_n(t))+X(t)]\) as \(n\to\infty\).
60F17 Functional limit theorems; invariance principles
60G10 Stationary stochastic processes