High-Water Mark

Given a sequence of values {a_k}_(k=1)^n, the high-water marks are the values at which the running maximum increases. For example, given a sequence (3,5,7,8,8,5,7,9,2,5) with running maxima (3,5,7,8,8,8,9,9,9), the high-water marks are (3,5,7,8,9), which occur at k=1, 2, 3, 4, and 8.


For independent random variables, the expected number of high-water marks after n measurements is H_n. 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 3!=6 possible orderings of measurements have the third as a record (so it contributes 2/3!=1/3), and so on (Havil 2003, pp. 125-126). A comparison of the number of records set in 10000 random trials of n measurements with H_n for n=1 to 100 is plotted above.

The number of records after n measurements is therefore |_H_n_|, which for n=1, 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 x records is therefore [n], where n are the values such that


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

See also

Local Maximum, Maximum, Running Maximum

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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."

Referenced on Wolfram|Alpha

High-Water Mark

Cite this as:

Weisstein, Eric W. "High-Water Mark." From MathWorld--A Wolfram Web Resource.

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