The Expected Number of Distinct Substrings in an Alphabet String

Abstract: Consider a sequence of i.i.d.~trials $X={X_1, X_2, \ldots, X_n}$ where $p(X_i=j)={1}/{d}; j=1,2,\ldots, d$, or more generally $p(X_i=j)=p_j; \sum_{1\le j\le d}p_j=1$. We consider the variable $D$ that counts the number of distinct substrings of all lengths,$1\le k\le n$ in $X$ and prove results concerning $E(D)$.