The asympt. to order 3 of continuous limit (as t goes to infinity) of the prob. density function for the normalized r.v. "number of inversions" of a*t 1's and b*t 2's up to order 3 in terms of the cont. variable x, and the symbols(!) a,b,t, is: / / 2 | | x 2 2 4 1/2 1/2 | 2 | exp(- ----) (a + a b + b ) x 2 Pi | x 1/2 1/2 | 2 1/2 |exp(- ----) 2 Pi + |-1/20 ----------------------------------------- | 2 \ a b (a + b) | | \ 2 x 2 2 1/2 1/2 exp(- ----) (a + a b + b ) 2 Pi 2 - 3/20 -------------------------------------- a b (a + b) 2 \ / x 2 2 2 1/2 1/2| | exp(- ----) (a + a b + b ) x 2 Pi | | 2 2 | | x + 3/10 -----------------------------------------|/t + |1/16800 exp(- ----) a b (a + b) / | 2 \ 4 2 2 2 2 2 2 3 2 3 2 4 x (-840 a b + 480 a x b + 320 x a b + 320 a x b + 160 x a 2 4 4 2 2 4 3 4 3 4 4 4 4 + 160 x b + 63 x b a + 42 x b a + 42 x b a + 21 x a + 21 x b ) 2 1/2 1/2 / 2 2 2 x 2 2 2 2 Pi / ((a + b) a b ) + 3/560 exp(- ----) (240 a x b / 2 2 2 4 3 4 3 4 2 2 2 3 - 28 a b + 98 x b a + 98 x b a + 147 x b a + 160 a x b 2 3 4 4 4 4 2 4 2 4 1/2 1/2 / + 160 x a b + 49 x b + 49 x a + 80 x b + 80 x a ) 2 Pi / / 2 2 2 2 x 2 2 2 2 2 2 ((a + b) a b ) - 1/1400 exp(- ----) x (600 a x b - 420 a b 2 4 3 4 3 4 2 2 2 3 2 3 + 98 x b a + 98 x b a + 147 x b a + 400 a x b + 400 x a b 4 4 4 4 2 4 2 4 1/2 1/2 / 2 2 + 49 x b + 49 x a + 200 x b + 200 x a ) 2 Pi / ((a + b) a / 2 x 2 2 3 3 4 4 1/2 1/2 exp(- ----) (3 a b + 2 a b + 2 a b + a + b ) 2 Pi 2 21 2 b ) + --- ------------------------------------------------------------ - 160 2 2 2 (a + b) a b 2 x 2 2 2 2 3 2 3 2 4 1/280 exp(- ----) (441 a x b + 294 x a b + 294 a x b + 147 x a 2 2 4 2 2 3 3 4 4 1/2 1/2 + 147 x b + 120 a b + 80 a b + 80 a b + 40 a + 40 b ) 2 Pi \ infinity | / /infinity | | | ----- / 2 2 2 | / 2 1 | | \ / / ((a + b) a b )| / t + ---- | | ) |-1/5250 x~ / | / 3 | | / \ / t | | ----- / \ x~ = 0 -infinity 6 4 6 3 6 2 3 (-1 + x~) (102 x~ + 102 x~ b + 7 x~ b - 18 x~ b + 300 b - 790 b 4 4 3 2 2 3 2 + 300 b + 7 x~ - 18 x~ - 43 x~ + 762 b x~ - 54 x~ b - 129 x~ b 4 5 5 3 2 2 4 2 3 + 21 x~ b + 306 x~ b + 306 x~ b - 391 b x~ + 42 b x~ - 108 b x~ 2 2 3 3 3 4 4 3 4 - 408 b x~ + 764 b x~ - 126 b x~ + 21 x~ b - 108 b x~ + 762 b x~ 3 4 3 5 2 5 4 2 4 4 2 6 + 49 b x~ - 54 x~ b - 129 x~ b - 408 b x~ + 42 b x~ - 43 x~ b ) \ | (2 x~) / x~ 3 3 \| 2 3 8 6 (-I x) / (x~! 2 b (1 + b) )|| (1 + 3 b + 3 b + b ) (7 x b / /| | / 8 5 8 4 2 2 4 2 2 3 3 6 3 3 + 21 x a b + 42 x b a + 4800 x a b + 3200 x a b - 1120 x a b 8 5 8 4 2 8 3 3 2 4 2 3 3 + 21 x a b + 42 x a b + 49 x a b + 4800 x b a - 16800 a b 6 6 4 4 2 8 6 4 6 4 6 - 160 x a + 3720 x a b + 7 x a + 720 x b + 720 x a 6 4 2 4 5 6 5 6 4 2 6 5 - 960 x b a + 2160 x a b - 480 x a b - 960 x a b - 480 x a b 4 4 2 6 6 4 5 4 3 3 2 / + 3720 x b a - 160 x b + 2160 x a b + 4680 x a b ) exp(x I) / / 6 8 6 6 5 2 4 4 4 4 5 ((-160 x + 7 x b - 480 x b + 4800 x b + 3720 x b + 2160 x b 3 2 3 8 2 6 2 8 4 8 - 16800 b + 3200 x b + 42 x b - 960 x b + 42 x b + 7 x 4 2 4 6 2 2 4 3 4 6 6 + 3720 x b + 720 x b + 4800 x b + 4680 x b + 720 x - 160 x b 8 3 6 4 8 6 3 8 5 6 + 49 x b - 960 x b + 21 x b - 1120 x b + 21 x b - 480 x b \ | | 4 3 3 | + 2160 x b) a (a + b) ) dx|/Pi | | | / This all things too, 342.214, seconds of CPU time