Asymptotic properties of the factors of words over a finite alphabet
Let A be an alphabet of cardinality m, k(n) be a sequence of positive integers and w is an element of A* (vertical bar w vertical bar = k(n)). In this paper it is shown that if lim sup(n ->infinity) k(n)/ln n  infinity) k(n)/ln n > 1/ln m, then this property is not true. Also, if lim inf(n ->infinity) k(n)/ln n > 1/ln m, then almost all words of length n over A do not contain the factor w. Moreover, if lim(n ->infinity)(ln n - k(n) ln m) = alpha is an element of R, then lim sup(n ->infinity) vertical bar W(n, k(n), w, A)vertical bar/m(n)  infinity) vertical bar W(n, k(n), w, A)vertical bar/m(n) >= 1 - exp(-(1 - 1/m) exp (alpha)), where W(n, k(n), w, A) denotes the set of words of length n over A containing the factor w of length k(n).
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