A numerical characteristic of extreme values

A numerical characteristic of large random numbers is studied. Let $F$ be a distribution on the real numbers with infinite endpoint. $X$ denotes a random variable with distribution $F$. Consider the transformation for a decimal number $d_1 d_2 d_3 \dots d_n.d_{n+1} \dots$ in $[10^{n-1} , 10^n)$ to $0.d_2 d_3 \dots$ in $[0, 1)$. We are interested in the distribution of transformed $X$ for large $X$, which implies the behavior of the large random number except the first figure. It is shown that the distribution of transformed $X$ conditioned by the first figure converges as $X$ becomes large for most distributions. Moreover, it turns out that the limit distribution depends on the tail behavior of $F$ and the first figure. A similar problem for distributions with finite endpoints is also considered. In this case, the distance until the endpoint is a matter of concern and parallel results to the ones for infinite endpoint case are given.

Keywords

random number, regular variation, Π-variation

2010 Mathematics Subject Classification

Primary 60E05. Secondary 60Fxx.

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Published 9 September 2014