An application of the Gaussian correlation inequality to the small deviations for a Kolmogorov diffusion. (English) Zbl 1490.60069

A Kolmogorov diffusion \((X_t)_{t\in[0,T]}\) is the stochastic process \(X_t=(X_1(t),\ldots,X_d(t))\) on \(\mathbb R^d\) whose marginals are iterated integrated Brownian motions given recursively by \(X_1(t)=B_t\) and \(X_{k+1}(t)=\int_0^t X_k(s)\,ds\), where \((B_t)_{t\geq0}\) is a standard one-dimensional Brownian motion. The author proves by an application of the Gaussian correlation inequality that the Kolmogorov diffusion fulfills the same small deviation principle as \(d\)-dimensional Brownian motion. As a consequence, Chung’s law of the iterated logarithm as \(T\to\infty\) and \(T\downarrow 0\) are provided for Kolmogorov diffusion. It turns out that as \(T\downarrow0\) the rate and constant in Chung’s law are the same as for \(d\)-dimensional Brownian motion, whereas rate and constant coincide with those of the iterated integrated Brownian motion \(X_d(t)\) as \(T\to\infty\).


60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems
60G15 Gaussian processes
60J65 Brownian motion
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