Brownian Motion Area with Generatingfunctionology

Transcription

1 Brownian Motion Area with Generatingfunctionology Uwe Schwerdtfeger RMIT University/University of Melbourne Alexander von Humboldt Foundation Monash University 3 August, 2011

2 Some continuous time processes... A Brownian Motion of duration 1 is a stochastic process B(t), t [0, 1] such that t B(t) is a.s. continuous, B(0) = 0, for s < t, B(t) B(s) N (0, t s) and increments are independent. A Brownian Meander M(t), t [0, 1] is a BM B(t) conditioned on B(s) 0, s ]0, 1]. A Brownian Excursion E(t), t [0, 1] is M(t) conditioned on M(1) = 0 (quick and dirty def.).

3 ... and their discrete counterparts The Bernoulli Random Walk B n (k) on Z, k {0, 1,..., n}, with B n (0) = 0, B n (k + 1) B n (k) { 1, 1}, each with prob. 1/2. The Bernoulli Meander M n (k), k {0,..., n} on Z 0 is B n (k) conditioned to stay non-negative. The Bernoulli Excursion E 2n (k), k {0,..., 2n} on Z 0 is M 2n (k) conditioned on M 2n (2n) = 0.

4 Scaling limits For n we have the weak limits { } n 1 B n ( nt ), t [0, 1] {B(t), t [0, 1]}, { } n 1 M n ( nt ), t [0, 1] {M(t), t [0, 1]}, { } 1 2n E 2n ( 2nt ), t [0, 1] {E(t), t [0, 1]}.

5 Drmota (2003): Weak limits imply moment convergence for certain functionals. E.g. for area (i.e. integrals) [ ( 1 ) r ] 1 E E 2n ( 2nt )dt E [EA r ], 2n E 0 [ ( 1 for n, where 0 EA := ) r ] 1 M n ( nt )dt E [MA r ], n 1 0 E(t)dt, MA := 1 0 M(t)dt. So studying functionals on E or M amounts to studying the discrete models!

6 Particularly EA appears in a number of discrete contexts, e.g. Construction costs of hash tables, cost of breadth first search traversal of a random tree, path lengths in random trees, area of polyominoes, enumeration of connected graphs. Many of the discrete results rely on recursions for the moments of EA and MA found by Takács (1991,1995) studying E 2n and M n.

7 Results We choose a different combinatorial approach and obtain new formulae for E (EA r ) and E (MA r ), the joint distribution of (MA, M(1)) in terms of the joint moments E (MA r M(1) s ), the joint distribution of (signed) areas and endpoint of B, and as an application of these area of discrete meanders with arbitrary finite step sets, area distribution of column convex polyominoes.

8 In the discrete world, we can write the joint distribution of the random variables as A n = n M n (k) and H n = M n (n) k=0 P(A n = k, H n = l) = p n,k,l r,s p, n,r,s where p n,k,l is the number of meanders of length n, area k and final height l.

9 The generating function of the class of meanders is the formal power series M(z, q, u) = p n,k,l q k u l z n, n k,l The above probabilities can be rewritten as P(A n = k, H n = l) = p n,k,l r,s p n,r,s [ z n q k u l] M(z, q, u) = [z n. ] M(z, 1, 1)

10 M(z, q, u) = n k,l p n,k,l q k u l z n, and [ z n q k u l] M(z, q, u) P(A n = k, H n = l) = [z n. ] M(z, 1, 1) With this representation the moments take a particularly nice form: E (A r nhn) s = k r l s P(A n = k, H n = l) k,l ( ) r (u [z n ] q ) s q u M(z, 1, 1) = [z n. ] M(z, 1, 1) So: large n behaviour of the moments by coefficient asymptotics of the above series.

11 Singularity analysis (Flajolet, Odlyzko 1990) Transfer Theorem: Let F (z) = f n z n be analytic in an indented disk and F (z) (1 µz) α (z 1/µ). Then f n [z n ] (1 µz) α 1 Γ(α) nα 1 µ n (n ). For example, it turns out, that ( ) r ( ) s b r,s M(z, 1, 1) q u (1 2z) 3r/2+s/2+1/2 (z 1/2),

12 Functional equation for M(z, q, u). The recursive description of the set of meanders translates into {meanders of length n} {meanders of length n 1} {, } \ {excursions of length n 1} { } M(z, q, u) = 1 + M(z, q, uq) E(z, q) is the generating function of excursions. ( zuq + z ) E(z, q) z uq uq,

13 Solution to the equation for q = 1 by the kernel method: z(u u 1 (z))(u v 1 (z))m(z, 1, u) = u ze(z, 1). where u 1 (z) = 1 1 4z 2 2z and v 1 (z) = z 2 2z. Substitution of u = u 1 (z) yields E(z, 1) = u 1(z) z = 1 1 4z 2 2z 2, and finally M(z, 1, u) = 1 z(u v 1 (z)).

14 The partial derivatives ( ) r ( ) s q u M(z, 1, u) can in principle be obtained inductively by taking derivatives of the functional equation (and setting q = 1). Each derivative w.r.t. u produces one new unknown function ( ) r ( ) s+1 q u M(z, 1, u). Each derivative w.r.t. q produces two new unknowns, ( ) r+1 ( ) r+1 q E(z, 1) and ( ) s q u M(z, 1, u) and hence requires another application of the kernel method.

15 The exact expressions for ( ) r ( ) s q u M(z, 1, u) and for ( ) r (u [z n ] q ) s E (A r nhn) s q u M(z, 1, 1) = [z n. ] M(z, 1, 1) are getting intractable. But we can keep track of the singular behaviour of ( ) r ( ) s ( ) r q u M(z, 1, 1) and ( ) s q u M(z, 1, u1 (z)) and via singularity analysis large n asymptotics for the moments.

16 One proceeds in two steps: First show by induction ( ) r ( ) s a r,s M(z, 1, u 1 (z)) q u (1 2z) 3r/2+s/2+1/2 (z 1/2), where a r,s = a r,s 1 + (s + 2)a r 1,s+2, and then by induction ( ) r ( ) s b r,s M(z, 1, 1) q u (1 2z) 3r/2+s/2+1/2 (z 1/2), where b r,s = b r,s 2 + (s + 1)b r 1,s+1, (s 1), b r,0 = b r 1,1 + a r 1,1.

17 Application of the transfer theorem finally yields: E (A r nhn) s b r,s Γ(1/2) b 0,0 Γ((3r + s)/2) n(3r+s)/2, and hence (after rescaling n 3/2 A n and n 1/2 H n ) b r,s is essentially E (MA r M(1) s ), similarly a r 1,1 is essentially E (EA r ).

18 Discrete meanders and excursions with arbitrary finite step sets: No result on convergence to M resp. E! But: Generating function satisfies a similar functional equation. Area moments for meanders and excursions can be computed in the same fashion, and are expressed in terms of the very same b r,s resp. a r 1,1! Result depends on the sign of the drift = mean of the step set.

19 Column convex polyominoes: Area distribution on polyominoes with fixed perimeter n. Similar functional equation as above. Similar arguments yield an EA limit law as n.

20 Acknowledgement Thank you for your attention! And thanks to Alexander von Humboldt Foundation RMIT University The ARC Centre of Excellence Mathematics and Statistics of Complex Systems (MASCOS) for financial support.

21 Ouch! Taking derivatives of the fct. eq. w.r.t. q and u allows recursive computation of F n,t. c 1 (1 zs(u))f n,0 (u)+z r i (u)g (n) i i=0 +zs(u) +z n t=2 n n l l=1 t=0 c 1 z n i=0 l=1 = zs(u)nf n 1,1 (u) ( ) n F n t,t (u) t ( n l )( n l t ( ) n u l r (l) (u)g (n l) i. l ) u l+t S (l) (u)f n l t,t (u)