Path properties of the primitives of a Brownian motion (Q2726653)

For a positive integer \(m\) consider a Gaussian process NEWLINE\[NEWLINE X_m(t) = \frac{1}{m!} \int_0^t (t-s)^m dW(s),\quad t \geq 0, NEWLINE\]NEWLINE where \(\{W(t)\), \(t \geq 0\}\) is a standard Brownian motion. The upper and lower a.s. bounds are given for the moduli of continuity of \(X_m(\cdot)\). Large increments of \(X_m(\cdot)\) are studied as well. The auxiliary exponential inequalities for supremum of appropriately normalized increments of \(X_m(\cdot)\) are established. Previously some asymptotical properties of \(X_m(\cdot)\) were investigated by \textit{A. Lachal} [e.g. Ann. Probab. 25, No. 4, 1712-1734 (1997; Zbl 0903.60071)] and \textit{H. Watanabe} [Trans. Am. Math. Soc. 148, 233-248 (1970; Zbl 0214.16502)].