Consider the random variables number of occurences of a factor (consecutive subword) in a random n-letter word in the alphabet {1,2},of 1121 and 2212 The average and variance for 1121 are n 39 13 n [- 3/16 + ----, - --- + ----] 16 256 256 The average and variance for 2212 are n 39 13 n [- 3/16 + ----, - --- + ----] 16 256 256 12 The correlation is, - 7/13 + ---- 13 n The asymptotics for the 8^2 mixed moments are 1/2 1/2 1/2 1/2 63 13 256 213 13 256 --------------- + ---------------- 1352 2704 n 22 [[1, 0, 1, ----------------------------------, 3 + -----, 1/2 169 n n 1/2 1/2 1/2 1/2 315 13 256 4635 13 256 ---------------- - ----------------- 676 17576 n 163050 ------------------------------------, 15 + ------, 1/2 2197 n n 1/2 1/2 1/2 1/2 6615 13 256 6213375 13 256 ----------------- - -------------------- 1352 456976 n 4505340 12 ----------------------------------------, 105 + -------], [0, - 7/13 + ----, 1/2 2197 n 13 n n 1/2 1/2 1/2 1/2 21 13 256 3 13 256 - --------------- - -------------- 1352 208 n 21 1178 ----------------------------------, - -- + -----, 1/2 13 169 n n 1/2 1/2 1/2 1/2 1701 13 256 11529 13 256 - ----------------- + ------------------ 8788 17576 n 105 5310 ----------------------------------------, - --- + -----, 1/2 13 169 n n 1/2 1/2 1/2 1/2 38745 13 256 8120565 13 256 - ------------------ + -------------------- 17576 456976 n 735 7069440 -------------------------------------------, - --- - -------, 1/2 13 28561 n n 1/2 1/2 1/2 1/2 121275 13 256 477358875 13 256 - ------------------- + ---------------------- 4394 1485172 n ----------------------------------------------], [1, 1/2 n 1/2 1/2 1/2 1/2 21 13 256 3 13 256 - --------------- - -------------- 1352 208 n 267 1370 ----------------------------------, --- - -----, 1/2 169 169 n n 1/2 1/2 1/2 1/2 441 13 256 4629 13 256 ---------------- - ----------------- 8788 17576 n 1095 156918 ------------------------------------, ---- - ------, 1/2 169 2197 n n 1/2 1/2 1/2 1/2 229635 13 256 5524755 13 256 ------------------- - -------------------- 228488 456976 n 6945 13638660 ------------------------------------------, ---- - --------, 1/2 169 28561 n n 1/2 1/2 1/2 1/2 405720 13 256 100109835 13 256 ------------------- - ---------------------- 28561 371293 n 58905 223968780 --------------------------------------------, ----- - ---------], [ 1/2 169 371293 n n 1/2 1/2 1/2 1/2 63 13 256 213 13 256 --------------- + ---------------- 1352 2704 n 21 1178 ----------------------------------, - -- + -----, 1/2 13 169 n n 1/2 1/2 1/2 1/2 441 13 256 4629 13 256 ---------------- - ----------------- 8788 17576 n 12705 181110 ------------------------------------, - ----- + ------, 1/2 2197 2197 n n 1/2 1/2 1/2 1/2 68985 13 256 1923561 13 256 - ------------------ + -------------------- 228488 456976 n 73815 21662880 -------------------------------------------, - ----- + --------, 1/2 2197 28561 n n 1/2 1/2 1/2 1/2 2584575 13 256 126020565 13 256 - -------------------- + ---------------------- 371293 742586 n 588735 2340213120 -----------------------------------------------, - ------ + ----------, 1/2 2197 371293 n n 1/2 1/2 1/2 1/2 342544545 13 256 300904116345 13 256 - ---------------------- + ------------------------- 2970344 77228944 n 22 ----------------------------------------------------], [3 + -----, 1/2 169 n n 1/2 1/2 1/2 1/2 1701 13 256 11529 13 256 - ----------------- + ------------------ 8788 17576 n 1095 156918 ----------------------------------------, ---- - ------, 1/2 169 2197 n n 1/2 1/2 1/2 1/2 68985 13 256 1923561 13 256 - ------------------ + -------------------- 228488 456976 n 910905 23765748 -------------------------------------------, ------ - --------, 1/2 28561 28561 n n 1/2 1/2 1/2 1/2 1550115 13 256 84777795 13 256 -------------------- - --------------------- 742586 1485172 n 6621345 3220734060 --------------------------------------------, ------- - ----------, 1/2 28561 371293 n n 1/2 1/2 1/2 1/2 2259134955 13 256 183619138815 13 256 ----------------------- - ------------------------- 38614472 77228944 n ---------------------------------------------------, 1/2 n 1/2 1/2 1/2 1/2 315 13 256 4635 13 256 ---------------- - ----------------- 62833995 415297134630 676 17576 n -------- - ------------], [------------------------------------, 28561 4826809 n 1/2 n 1/2 1/2 1/2 1/2 229635 13 256 5524755 13 256 ------------------- - -------------------- 105 5310 228488 456976 n - --- + -----, ------------------------------------------, 13 169 n 1/2 n 1/2 1/2 1/2 1/2 1550115 13 256 84777795 13 256 -------------------- - --------------------- 73815 21662880 742586 1485172 n - ----- + --------, --------------------------------------------, 2197 28561 n 1/2 n 81780615 3462166800 - -------- + ----------, 371293 371293 n 1/2 1/2 1/2 1/2 695241225 13 256 61344411525 13 256 - ---------------------- + ------------------------ 38614472 77228944 n ---------------------------------------------------, 1/2 n 722430765 529070323950 - --------- + ------------, 371293 4826809 n 1/2 1/2 1/2 1/2 146452587435 13 256 17770706799615 13 256 - ------------------------- + --------------------------- 250994068 501988136 n 163050 ---------------------------------------------------------], [15 + ------, 1/2 2197 n n 1/2 1/2 1/2 1/2 38745 13 256 8120565 13 256 - ------------------ + -------------------- 17576 456976 n 6945 13638660 -------------------------------------------, ---- - --------, 1/2 169 28561 n n 1/2 1/2 1/2 1/2 2584575 13 256 126020565 13 256 - -------------------- + ---------------------- 371293 742586 n 6621345 3220734060 -----------------------------------------------, ------- - ----------, 1/2 28561 371293 n n 1/2 1/2 1/2 1/2 695241225 13 256 61344411525 13 256 - ---------------------- + ------------------------ 38614472 77228944 n ---------------------------------------------------, 1/2 n 9029822355 561077158230 ---------- - ------------, 4826809 4826809 n 1/2 1/2 1/2 1/2 45513138195 13 256 5909813510895 13 256 ------------------------ - -------------------------- 250994068 501988136 n -----------------------------------------------------, 1/2 n 1/2 1/2 1/2 1/2 6615 13 256 6213375 13 256 ----------------- - -------------------- 93550848615 96036178718970 1352 456976 n ----------- - --------------], [----------------------------------------, 4826809 62748517 n 1/2 n 1/2 1/2 1/2 1/2 405720 13 256 100109835 13 256 ------------------- - ---------------------- 735 7069440 28561 371293 n - --- - -------, --------------------------------------------, 13 28561 n 1/2 n 1/2 1/2 1/2 1/2 2259134955 13 256 183619138815 13 256 ----------------------- - ------------------------- 588735 2340213120 38614472 77228944 n - ------ + ----------, ---------------------------------------------------, 2197 371293 n 1/2 n 722430765 529070323950 - --------- + ------------, 371293 4826809 n 1/2 1/2 1/2 1/2 45513138195 13 256 5909813510895 13 256 ------------------------ - -------------------------- 250994068 501988136 n -----------------------------------------------------, 1/2 n 1175006091105 100904547110850 - ------------- + ---------------, 62748517 62748517 n 1/2 1/2 1/2 1/2 13751974746405 13 256 2460601591088445 13 256 - --------------------------- + ----------------------------- 6525845768 13051691536 n -------------------------------------------------------------], [ 1/2 n 1/2 1/2 1/2 1/2 121275 13 256 477358875 13 256 - ------------------- + ---------------------- 4505340 4394 1485172 n 105 + -------, ----------------------------------------------, 2197 n 1/2 n 1/2 1/2 1/2 1/2 342544545 13 256 300904116345 13 256 - ---------------------- + ------------------------- 58905 223968780 2970344 77228944 n ----- - ---------, ----------------------------------------------------, 169 371293 n 1/2 n 62833995 415297134630 -------- - ------------, 28561 4826809 n 1/2 1/2 1/2 1/2 146452587435 13 256 17770706799615 13 256 - ------------------------- + --------------------------- 250994068 501988136 n ---------------------------------------------------------, 1/2 n 93550848615 96036178718970 ----------- - --------------, 4826809 62748517 n 1/2 1/2 1/2 1/2 13751974746405 13 256 2460601591088445 13 256 - --------------------------- + ----------------------------- 6525845768 13051691536 n -------------------------------------------------------------, 1/2 n 176470995013425 20010566463237000 --------------- - -----------------]] 815730721 815730721 n This took, 456.779, seconds.