High-Water Mark

Given a sequence of values , the high-water marks are the values at which the running maximum increases. For example, given a sequence  with running maxima , the high-water marks are , which occur at , 2, 3, 4, and 8.

For independent random variables, the expected number of high-water marks after  measurements is . This can be seen by noting that the first measurement must by definition be a record (so it contributes 1), the second measurement is equally likely to be higher or lower than the first (so it contributes 1/2), two of the  possible orderings of measurements have the third as a record (so it contributes ), and so on (Havil 2003, pp. 125-126). A comparison of the number of records set in  random trials of  measurements with  for  to 100 is plotted above.

The number of records after  measurements is therefore , which for , 2, ... is given by 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, ... (OEIS A055980). The number of measurements needed to obtain  records is therefore , where  are the values such that

giving for , 2, 3, ... the values 1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, ... (OEIS A004080), and for , 10, 100, ... records are therefore 1, 12367, 15092688622113788323693563264538101449859497, ... (OEIS A096618).

REFERENCES:

Havil, J. "Setting Records." §13.4 in Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 125-126, 2003.

Sloane, N. J. A. Sequences A004080, A055980, and A096618 in "The On-Line Encyclopedia of Integer Sequences."