I used Marsaglia's 64-bit SUPER KISS pseudo-random number generator to simulate uniform r.v.s on [0, 1] that are independent, as X_1, X_2, X_3, ad infinitum
For each go, (or sequence) I define its 1st record-breaking value as R(1) as X_1, its 2nd record-breaking value R(2) as the value taken by X_n for the smallest n with X_n > X_1, and in general R(k+1) as the value taken by the smallest n with X_n > R(k), for k = 2, 3, 4, 5, ...
In my first simulation I get: R(20) = 0.999999999945556 or about 5.4E-11 less than 1 , a one in 18 billion event.
In fact, R(20) is about 1 - (0.307)^20 ...
So, I'm wondering about the asymptotics of 1 - R(k) for very large k. Of course, R(k) is a andom variable with a probability distribution. Can we say something about the asymptotics of 1 - R(k) for large k?
-- dracut:/# lvm vgcfgrestore File descriptor 9 (/.console_lock) leaked on lvm invocation. Parent PID 993: sh Please specify a *single* volume group to restore.