Where´s the peak? [causal signal with average delay]

Two results are derived concerning the peak (i.e., maximum amplitude) of a causal signal with a given average delay. The first result is that, for an average delay of τ, the maximum possible location of the signal peak is on the order of τ(τ+3)/2. (This bound can also be interpreted as providing the maximum integer at which the most probable value of a discrete nonnegative random variable could occur, given that the random variable has a known mean.) The second result is that the signals that minimize the peak amplitude, subject to unit energy and average delay τ, have a peak value of the order of 1/√(2τ+1). Causal signals for which the derived bounds are attained for any given real-valued delay are constructed. The derived bounds are compared to the corresponding ones for all-pass signals