8 Laplace s Method and Local Limit Theorems


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1 8 Lplce s Method nd Locl Limit Theorems 8. Fourier Anlysis in Higher DImensions Most of the theorems of Fourier nlysis tht we hve proved hve nturl generliztions to higher dimensions, nd these cn be proved either by mimicking the onedimensionl rguments or by deducing them directly from the onedimensionl results with the help of the Fubini theorem. Here we will look only t rndom vectors tht tke vlues in the integer lttice k, with the im of developing n inversion formul tht will llow us to settle the question of recurrence/trnsience for k dimensionl rndom wlk. Definition 8.. The chrcteristic function of k dimensionl rndom vector X is the function ' X : R d! C defined by ' X (µ) = Ee ihµ,x i where hu, vi denotes the inner product of the vectors u, v. If the rndom vector X tkes vlues in k then the chrcteristic function ' X (µ) is periodic in ech coordinte µ j. Lemm 8.2. (Orthogonlity Reltions) The exponentil functions e ihm,µi re orthonorml in L 2 [ º,º] k, tht is, for ny two elements m,n 2 k, () k e ihm,µi e ihn,µi dµ dµ 2 dµ k = ± m,n. (8.) [ º,º] k Corollry 8.3. If X tkes vlues in k then P(X = m) = () k ' X (µ)e ihm,µi dµ dµ 2 dµ k. (8.2) [ º,º] k For simple rndom wlk S n on k, the step distribution hs chrcteristic function '(µ) = k P k j = cosµ j ; therefore, the return probbilities re given by P(S n = 0) = () k [ º,º] k k kx cosµ j dµ dµ 2 dµ k. (8.3) j = 8.2 Lplce s Method The first cse of the centrl limit theorem, for sumes of independent, identiclly distributed Bernoulli rndom vribles, ws proved in the 730s by De Moivre. De Moivre proof relied on Stirling s formul n! ª n n e np n nd the fct tht the binomil distribution cn be explicitly written in terms of fctorils. Some 40 (?) yers lter Lplce provided new nd better pproch, using wht is now clled the Lplce method 79
2 of symptotic expnsion. This is fundmentlly importnt technique in mthemticl nlysis, nd one which is especilly useful in probbility nd sttistics, nd it is the esiest route to the locl centrl limit theorem. It relies on the following fct, which I will tke s known. Lemm 8.4. e x2 /2 dx =. The Lplce method is technique for obtining shrp pproximtions to integrls of the form (or similr in form to) e ng(x) dx (8.4) where g is smooth relvlued function tht ttins its mx in [,b] uniquely t n interior point, which I will ssume (without loss of generlity) is the origin. Since g is mximl t x = 0, its first derivtive is g 0 (0) = 0 nd its second derivtive is nonpositive. There is no loss of generlity in ssuming tht g (0) = 0 (becuse subtrcting constnt from g just multiplies the entire integrl by e n ). Lplce s gret insight ws tht if in ddition the second derivtive g 00 (0) < 0 is nonzero, then when n is lrge the integrnd e ng(x) spikes very shrply t 0, nd in neighborhood of 0 of width 3/ p n or 5/ p n or so the twoterm Tylor series pproximtion to g gives e ng(x) º e ng00 (0)x 2 /2, so the spike looks very much like Gussin distribution. Formlly replcing the integrnd by this Gussin distribution nd integrting (using Lemm 8.4 together with the substitution y = p nx) then yields the pproximtion p e ng(x) dx º e ng00 (0)x 2 /2 dx = p n( g 00 (0)). Theorem 8.5. (Lplce) Assume tht g is smooth (t lest C 2 ) function on the intervl [,b], where < 0 < b, nd ssume tht () g (0) = 0; (b) g 00 (0) < 0; nd (c) g (x) < 0 for every x 2 [,b] except x = 0. Then s n!, e ng(x) dx ª p p ng 00 (0). (8.5) The symbol ª mens tht the limit of the rtio of the two sides converges to s n! (equivlently, the reltive error in the pproximtion goes to 0). 80
3 Proof of Theorem 8.5. The integrl J n will be nlyzed by breking the rnge of integrtion into two prts: (i) smll intervl [ ±,±] contining 0, nd (ii) everything else, tht is, [, ±] [ [±,b]. The second region (ii) is esier, so we begin with this. By hypothesis, the function g is strictly less thn 0 t every x 6= 0, nd it is continuous on the intervls [, ±] nd [±,b], so there exists = (±) > 0 such tht g (x) for every x 2 [, ±] [ [±,b]. Consequently, ± e ng(x) dx+ ± ng(x) dx e n (b ). Since e n is negligible compred to n /2 s n!, to complete the proof it will suffice to show tht for ny " > 0 there is ± > 0 such tht for ll lrge n, " < p ±.q n e ng(x) dx /( g 00 (0)) < + ". ± Now for ny " > 0 there exists ± > 0 such tht in the intervl [ ±,±] the twoterm Tylor series pproximtion to g (x) is ccurte to within fctor (±") 2, tht is, for ll x 2 [ ±,±], ( ") 2 g 00 (0)x 2 /2 < g (x) < ( + ") 2 g 00 (0)x 2 /2. Hence, for x 2 [ ±,±] the integrnd e ng(x) is bounded bove nd below by Gussin densities with vrinces (( ± ") 2 /( g 00 (0)n). Consequently, the integrl R ± ± eng(x) dx is bounded bove nd below by the integrls of the bounding Gussin densities, which by Lemm 8.4 re s ( ± ") g 00 (0)n. (Note: becuse the Gussin densities hve vrinces of order /n, the totl mss outside [ ±, ±] is exponentilly smll, by Homework 6, problem 3.) We hve lredy encountered clss of integrls tht very similr in form to (8.4), in the Fourier inversion formul (??). In generl, the chrcteristic functions ' X (µ) of integervlued rndom vribles X re not relvlued, nd furthermore the integrls (??) contin n dditionl fctor e imµ, so 8.5 does not pply directly. But there re some importnt specil cses where Lplce s theorem does pply directly. Exmple 8.6. Let X, X 2,... be independent, identiclly distributed Rdemcher rndom vribles with prtil sums S n. As we hve seen, the chrcteristic function of S n is ' Sn (µ) = cos n µ. Clerly, when n is odd the return probbility P{S n = 0} = 0. When n is even, the chrcteristic function cos n µ hs period º, nd so the Fourier inversion formul (??) implies 8
4 tht Hence, by Lplce s theorem, P{S n = 0} = = º º º º/2 º/2 P{S 2n = 0} ª cos n µ dµ cos n µ dµ r ºn. Exmple 8.7. Let {p n } n2e symmetric probbility distribution on the integers (tht is, p n = p n ) with vrince 0 < æ 2 <, nd let X, X 2,... be independent, identiclly distributed with distribution P{X n = m} = p m. Then the chrcteristic function '(µ) = Ee iµx is relvlued nd hs even, nd hs second derivtive ' 00 (0) = æ 2. Assume tht the support of the distribution {p n } n2 is not contined in n rithmetic progression m + b where 2. Then by Proposition??, '(µ) < for every µ 2 [ º,º] except µ = 0. By the Fourier inversion formul (??), P{S n = 0} = º '(µ) n dµ. º Since ' is relvlued nd hs unique mx t µ = 0, Lplce s theorem pplies, so we conclude tht P{S n = 0} ª p. næ Exercise 8.8. () Show tht n! = R 0 xn e x dx. (b) Deduce tht n! = n n+ e n R 0 yn e (y )n dy. (c) Adpt the proof of Lplce to show tht n! ª n n e np n. Lplce s method extends without gret difficulty to multiple integrls. The only differences with the onedimensionl cse re (i) one must replce the use of Tylor s pproximtion to g (x) ner the origin by multivrite Tylor pproximtion; nd (ii) the Gussin densities tht bound e ng(x) for x ner the origin must be replced by multivrite Gussin densities. Following is prticulr cse of interest, where the relevnt multivrite Gussin density is just the product of onedimensionl Gussin densities. Theorem 8.9. Let g :[, ] d! R be smooth function tht stisfies the following hypotheses: () g (0) = 0; (b) g (x) < 0 for ll x 6= 0; (c) D 2 g (0) = æ 2 I for some 0 < æ 2 <, 82
5 where D 2 g is the d d mtrix of second prtil derivtives nd I is the d d identity mtrix. Then s n!, µ d/2 e ng(x) dx ª æ d. (8.6) [,] d n Exmple 8.0. Let S n be the loction of simple nerest neighbor rndom wlk on the d dimensionl integer lttice fter n steps. Then µ d d/2 P{S 2n = 0} ª ºn Consequently, the expected number of returns P n= P{S 2n = 0} to the origin is finite if d 3, Thus simple rndom wlk in 3 dimensions nd higher is trnsient. 83
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