three steps analogously to Lemma 6, and we consider two subcases, sequences that terminate in a 1 (i.e. a run of 1s of length less than k) and sequences that terminate in a (i.e. a run of 0s). For those sequence that terminate in a 1, for a given (r11, . . . , r1k−1, s1k ) the total number of sequences satisfying this criteria is: f1k 1 s1k r1! − n0 C1k (r1 1)! = r1 − − − k−1 r1j ! j=1 j=1 1j s1k 1 − r1 1 − (s1k 1)! − n0 + 1 with (r1 1)!/((s1k 1)! r1j !) being the number of sequences that terminate in a run of 1s − − k−1 of length k or more. For those sequences that terminate in a 0, for a given (r11, . . . , r1k−1, s1k ) the total number of sequences satisfying this criteria is: 1k r1! n0 + 1 − n0 f1k 1 r1 − n0 + 1 s1k 1 r1j ! r1 − j=1 therefore, the sum of the k/1-streak momentum statistic across all sequences for which it is defined satisfies: n l ) s1k − f1k s1k + 1 ,s1k jr1j <n1 1 j=1 −k E[P1k = n1] 0 |N1 f1k 1 s1k − ≥1 = C1k U1k − r11 ,...,r1k−1 + ) C1k r1 s1k f1k s1k − − f1k r11 ,...,r1k−1 ,s1k j=1 jr1j n0 + 1 ≤n1 −k s1k ≥1 + ) C1k
three steps analogously to Lemma 6, and we consider two subcases, sequences that terminate in a 1 (i.e. a run of 1s of length less than k) and sequences that terminate in a (i.e. a run of 0s). For those sequence that terminate in a 1, for a given (r11, . . . , r1k−1, s1k ) the total number of sequences satisfying this criteria is: r1! (r1 − 1)! n0 f1k − 1 = r1 − s1k C1k − j=1 r1j ! (s1k − 1)! k−1 j=1 1j r1 − 1 s1k − 1 n0 + 1 with (r1 − 1)!/((s1k − 1)! k−1 r1j !) being the number of sequences that terminate in a run of 1s of length k or more. For those sequences that terminate in a 0, for a given (r11, . . . , r1k−1, s1k ) the total number of sequences satisfying this criteria is: r1! n0 f1k − 1 n0 + 1 − r1 1k j=1 r1j ! r1 s1k − 1 n0 + 1 therefore, the sum of the k/1-streak momentum statistic across all sequences for which it is defined satisfies: n l ) s1k f1k − s1k E[P1k |N1 = n1] − U1k 1 = r11 ,...,r1k−1 ,s1k j=1 jr1j <n1 −k s1k ≥1 C1k 0 + 1 f1k − 1 + ) C1k r1 − s1k f1k − s1k r11 ,...,r1k−1 ,s1k j=1 jr1j ≤n1 −k s1k ≥1 n0 + 1 f1k + ) C1k
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