The law of iterated logarithm for \(m\)-dependent Banach space valued random variables (Q1366727)

The minimal conditions for \(m\)-dependent Banach space valued random elements satisfying the law of iterated logarithm (LIL) are concerned. Roughly speaking, it is proved that the classical conditions for an i.i.d. sequence to satisfy LIL are still sufficient for LIL in \(m\)-dependent case. Such result cannot be obtained in a way similar to what we did in the one-dimensional case (the appropriate example is given). Recall that a sequence \((X_n)_{n\geq 1}\) of strictly stationary random elements is called \(m\)-dependent, where \(m\) is nonegative integer, if for any \(k\geq 1\), the two collections \(\{X_1,\cdot , X_k\}\) and \(\{X_{k+m+1}, X_{k+m+2},\cdot \}\) are independent.