Roderick Wong. Asymptotic Approximations of Integrals. Academic Press, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in (2.18b) and (2.26) respectively. For such a ball domain, the gradient of R0 was calculated previously in [21] as V0 ix0[ x0. By symmetry, we need only look for an equilibrium solution to (3.1) on the segment x0 [0, 1) of the positive real axis. To do so, we need Laplace s formula (cf. [26]) valid for z 44 1, r F(r)e dS F(r) 1 nr) V e . 3.3a) Comparing (2.25b) with (3.3a) we take ( a.a) In (3.3a) r dist (x0; 0) n is the curvature of 0 at x, and the sum is taken over all x 0 that are closest to x0. The sign convention is such that n 0 if is convex ....

R. Wong, Asymptotic Approximations of Integrals, Academic Press, San Diego, CA, (1989). 37

....analytically inherent in the underlying Fourier integrals is a simple pole and a saddlepoint. From this perspective, the appearance of normal distribution in our problems is not so unexpected because of the appearance of a double pole and a moving saddlepoint in the integral representation (cf. [17, 2, 29]) In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. 1, 29] Recurrence relation. It is more convenient to work with E nk : Q ....

.... pole and a moving saddlepoint in the integral representation (cf. 17, 2, 29] In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. [1, 29]) Recurrence relation. It is more convenient to work with E nk : Q n0 nk , since E 0k = E 1k = E 2k = E n0 = 0. 7) Using the combinatorial identities j) j 1)H j = n 1)H n 12 (n 1) 31) 8) j) j 1) n 1 11) j) 2 #(3n ....

[Article contains additional citation context not shown here]

R. Wong, Asymptotic Approximations of Integrals, Academic Press, Inc., Boston, 1989. 24

.... in decomposable combinatorial structures (cf. 21] Thus we investigate the asymptotic behaviour of #m (#) as # and m runs through its possible values (depending on #) When # is bounded and the asymptotic behaviour of #m (#) can be easily derived by the usual saddlepoint method, cf. [10, 24, 26, 50]. For completeness, we include the resulting formula at the end of 2.2. Let us first list some known asymptotic estimates of #m (#) in the literature. They are not intended to be complete but are chosen according to the variation of the second parameter m. For more information on other types ....

....expansion (3) is useful. 3. In the ranges 0 # # or m # A # #, A 0, our results can be used. Many other types of normal approximation (usually of the form #m (#) #(g(#, m) can be found in [31, 2, 13] Concerning the case when m and # bounded, we have by the saddlepoint method (cf. [50]) e # m r m 1 # 2#m # 12# 12m 288# 888# 313 where r = m # 1. Note that this expansion can also be obtained from (3) but with more involved computations. 2.3 Poissonization Poissonization is a widely used technique in stochastic process, summability of divergent sequence, ....

R. Wong, Asymptotic approximations of integrals, Academic Press, Inc., Boston, 1989. Institute of Statistical Science Academia Sinica Taipei, 11529 Taiwan e-mail: hkhwang@stat.sinica.edu.tw 18

....A. Thus # (k 1)# 2 1 O k,# where x = m #) # #. Since the k (x) s are piecewise di#erentiable (except for at the zeros of h k (x) we can divide the summation range into k 1 subintervals and in each of which apply the Euler Maclaurin summation formula (cf. [94]) One may also apply the argument by Prohorov [73] In this way, the sum is well approximated by its integral counterpart and we obtain (13) Proof of (2) and (4) Set # m = P(X n = m) P(Y = m) As in the last section, we take A = # and set M 1 = # A # # and M 2 = # A # #. Consider ....

....(27) uniformly in each specified range of w. 20 Proof. By singularity analysis, we have ] 1 z) #w = n #w B n (w) where B n (w) n #w 2 , #. Note that Stirling s formula loses uniformity for parameter near the negative real axis, the so called Stokes phenomenon (cf. [94]) We decompose D n (w) into five parts: D n (w) # k (w) 1 # k (w)B n k (w) # #n 2# k#n = #(1, w) Z 0 (n) Z 1 (n) Z 2 (n) Z 3 (n) say. By hypothesis, # n (w) n 1 # , we have k (w) For Z 1 (n) if #w 1, we ....

Wong, R. (1989) Asymptotic approximations of integrals. Academic Press, Inc., Boston.

.... decomposable combinatorial structures (cf. 21] Thus we shall investigate the asymptotic behaviour of Pi m ( as 1 and m runs through its possible values (depending on ) When is bounded and m 1, the asymptotic behaviour of Pi m ( can be easily derived by the usual saddle point method, cf. [10, 24, 26, 50]. For completeness, we shall include the resulting formula at the end of x 2.2. Let us first list some known asymptotic estimates of Pi m ( in the literature. They are not intended to be complete but are chosen according to the variation of the second parameter m. For more information on other ....

....is useful. 3. In the ranges 0 m Gamma A or m A , A 0, our results can be used. Many other types of normal approximation (usually of the form Pi m ( Phi(g( m) can be found in [31, 2, 13] Concerning the case when m 1 and bounded, we have by the saddle point method (cf. [50]) 12 Gamma 13 12m 288 Gamma 888 313 where r = m= 1. Note that this expansion can also be obtained from (3) but with more involved computations. 2.3 Poissonization Poissonization is a widely used technique in stochastic process, summability of divergent sequence, ....

R. Wong, Asymptotic approximations of integrals, Academic Press, Inc., Boston, 1989.

....etc) for combinatorial distributions, the general intuition being that discrete approximations (as opposed to continuous ones) to discrete structures would usually provide better error estimates. Next, this line of study introduces many intriguing problems for uniform asymptotic analysis (cf. [7, 43]) and would be of special interest to analysts. Third, for combinatorialists and number theorists, the quantitative results demand further structural interpretations and characterizations. Problems like why it is Poisson for small m and negative binomial for large m will shed further light on ....

....like unimodality would also follow. Asymptotic behaviors of the two sequences [u , u for m log n can be treated by application of the singularity analysis and Selberg s method (cf. 9, 40, 26] Di#erent methods, like the two dimensional saddle point method (cf. [22, 43]) are, however, required when m log n. The methods used in this paper can also be applied to the number of irreducible factors (multiplicities counted) in the polynomial factorization of the characteristic polynomial in a random element in GL n (F q ) with bivariate generating function (cf. ....

R. Wong, Asymptotic approximations of integrals, Academic Press, Inc., Boston, 1989.

....of the convolution law of a Poisson and a negative binomial distributions. The analytic context encountered here is more complicated than the previous one (for M n;d ) and consists of a saddlepoint and a pole of order q in the integrand. Thus the appearance of D Gammaq (x) is quite expected; see [5, 31]. An intuitive interpretation of the result is that when Y n is large, most of the irreducible factors are of degree 1. Thus we can write Y n n Z n , where Y n and Z n count the number of irreducible factors of degree 2 and = 1, respectively, and prove that the Poisson behavior comes from ....

R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, MA (1989). 12

....analytically inherent in the underlying Fourier integrals is a simple pole and a saddlepoint. From this perspective, the appearance of normal distribution in our problems is not so unexpected because of the appearance of a double pole and a moving saddlepoint in the integral representation (cf. [17, 2, 29]) In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. 1, 29] Recurrence relation. It is more convenient to work with E nk : Q ....

.... pole and a moving saddlepoint in the integral representation (cf. 17, 2, 29] In general, when algebraic singularity and saddlepoint may coalesce in the integrand, good approximants are parabolic cylinder functions which are certain weighted integrals of the normal distribution function (cf. [1, 29]) Recurrence relation. It is more convenient to work with E nk : Q n0 Q nk ; since E 0k = E 1k = E 2k = E n0 = 0: 7) Using the combinatorial identities j(n 1 j) j 1)H j = n 1)H n 12 (n 1) 1 (8) j(n 1 j) j 1) n 1 j(n 1 j) 2 (3n 8) we obtain ....

[Article contains additional citation context not shown here]

R. Wong, Asymptotic Approximations of Integrals, Academic Press, Inc., Boston, 1989. 24

....etc) for combinatorial distributions, the general intuition being that discrete approximations (as opposed to continuous ones) to discrete structures would usually provide better error estimates. Next, this line of study introduces many intriguing problems for uniform asymptotic analysis (cf. [7, 43]) and would be of special interest to analysts. Third, for combinatorialists and number theorists, the quantitative results demand further structural interpretations and characterizations. Problems like why it is Poisson for small m and negative binomial for large m will shed further light on ....

....properties like unimodality would also follow. Asymptotic behaviors of the two sequences ; u for m log n can be treated by application of the singularity analysis and Selberg s method (cf. 9, 40, 26] Different methods, like the two dimensional saddle point method (cf. [22, 43]) are, however, required when m AE log n. The methods used in this paper can also be applied to the number of irreducible factors (multiplicities counted) in the polynomial factorization of the characteristic polynomial in a random element in GL n (F q ) with bivariate generating function (cf. ....

R. Wong, Asymptotic approximations of integrals, Academic Press, Inc., Boston, 1989.

.... Gamma(k 1)ff=2 1 O k;ff where x = m Gamma ) Since the jh k (x)j s are piecewise differentiable (except for at the zeros of h k (x) we can divide the summation range into k 1 subintervals and in each of which apply the Euler Maclaurin summation formula (cf. [94]) One may also apply the argument by Prohorov [73] In this way, the sum is well approximated by its integral counterpart and we obtain (13) Proof of (2) and (4) Set ffi m = P(X n = m) Gamma P(Y = m) As in the last section, we take A = and set M 1 = Gamma A and M 2 = A . ....

....uniformly in each specified range of w. 20 Proof. By singularity analysis, we have ] 1 Gamma z) n w Gamma 1 B n (w) B n (w) n uniformly for jwj j. Note that Stirling s formula loses uniformity for parameter near the negative real axis, the so called Stokes phenomenon (cf. [94]) We decompose D n (w) into five parts: D n (w) Xi k (w) Gamma Xi k (w) 1 Xi k (w)B n Gammak (w) bn=2c kn = Xi(1; w) Z 0 (n) Z 1 (n) Z 2 (n) Z 3 (n) say. By hypothesis, Xi n (w) n j Xi k (w)j n For Z 1 (n) if w 6= 1, we ....

Wong, R. (1989) Asymptotic approximations of integrals. Academic Press, Inc., Boston.

....terms of the convolution law of a Poisson and a negative binomial distributions. The analytic context encountered here is more complicated than the previous one (for M n,d ) and consists of a saddlepoint and a pole of order q in the integrand. Thus the appearance of D q (x) is quite expected; see [5, 31]. An intuitive interpretation of the result is that when Y n is large, most of the irreducible factors are of degree 1. Thus we can write Y n = Y # n Z n , where Y # n and Z n count the number of irreducible factors of degree 2 and = 1, respectively, and prove that the Poisson behavior comes ....

R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, MA (1989). 12

....2 gives access to successive lower order terms of the polynomials A k (t) near t = 0. Here, we develop a more sophisticated analysis based on a method of coalescent saddle points whose principles originate with Chester, Friedman, and Ursell [6] and which is exposed in classical treatises like [4, 25, 33]. In particular, we follow closely the treatment o ered by Olver in [25, p. 352 356] Proceeding along these lines, we establish below the existence of an expansion (31) Q(z; q) B k (t; 0 that is valid for t in a closed subinterval of (0; 1] for instance t 2 [ 4 ; 1] It will ....

Wong, R. Asymptotic approximations of integrals. Academic Press Inc., Boston, MA, 1989.

....in our abstract. Throughout this paper, A B means that A and B share the same leading term ( when expanded in terms of a certain asymptotic parameter ) The symbol a 1 means that a is small enough ( this usually can be characterized by some asymptotic parameter ) We refer to [1, 5, 7, 11] for a full theory of the asymptotic analysis of integrals. 2. ACCURACY OF APPROXIMATIONS We define the following approximations to f p ( t ) f O p (v)e dv (frequency limited to v p) 6) iarg ( f O p ) dv (magnitude taken as 1) 7) F p ( t ) dv (leading term of ....

....iii ) FisC around this critical point and F 9 (c) x 0 . Then the leading asymptotic magnitude is proportional to 1 l: f(c) lF9(c) e 0i[lF(c) sign ( F 0 ( c) p 4] o(l 01 2 ) 25) The proofs of these two statements can be found in many asymptotic analysis textbooks ( for instance [1, 5, 7, 11] ) with a little modification on the regularity of F . Next, let s consider the doubly parameterized Fourier integral (DPFI) e 0ilF(v,t) dv, for real t. 26) At any fixed time t, Statements 1 and 2 can be applied to DPFI. As long as the regularity conditions for v are satisfied uniformly ....

R. Wong, "Asymptotic Approximations of Integrals," Academic Press, New York, 1989.

....If that is, z s, phz ( r] or phz , ze 2i 3 are both inside , and we can use (see (1.5) At most one of the Airy functions at the right hand side is dominant if z . 2.2) 3. STEEPEST DESCENT CONTOURS For details on the saddle point method and steepest descent contours we refer to [10] or [13]. We consider Ai(z) c eW3 zw dw, 3.1) where ph z [0, r] and Co is the contour shown in Figure 1. w) w 3 zw. Let (3.2) The saddle points are wo = V and w = wo and follow from solving b(w) w 2 z = 0. The path of steepest descent through the saddle point wo is defined by [ w) ....

R. Wong (1989). Asymptotic approximations of integrals. Academic Press, New York.

.... we are able to improve on the complexity of the best known random samplers for multiply connected planar graphs and convex polyhedra from [44] The analysis that we introduce is largely based on a method of coalescing saddle points that was perfected in the 1950 s by applied mathematicians [3, 8, 52] and has found scattered applications in statistical physics and the study of phase transitions [41] However, this method does not appear to have been employed so far in the eld of random combinatorics. We claim some generality for the approach Date: August 31, 2001. Key words and phrases. ....

....phenomenon to be described is the coalescence of two simple saddle points into a double one 6 . We follow the book of Bleistein and Handelsman [8, Sec. 9.2] where the method originally due to Chester, Friedman, and Ursell is exposed (see also the books by Olver [39, pp. 351 361] and Wong [52]) The simplest occurrence of the phenomenon appears in the integration of exp(nf(t) with a cubic function f , f(t) t 3 3 2 t r: Indeed, in this case there are two saddle points and (given by f 0 (t) t 2 2 ) coalescing into a double saddle point as 0. The ....

Wong, R. Asymptotic Approximations of Integrals. Academic Press, 1989.

....the inverse Fourier transform to both sides of Eq. 24 gives the integral representation of the Bessel function ( 1, Eq. 9.1.21] J (z) 1= 2) Z Gamma e i(z sin Gamma ) d : 25) From Eq. 25 one can derive the behavior of J (z) for j j z by employing the stationary phase method [9]. It can thus be seen that for values of with j j z one must expect jJ (z)j to be relatively large since z sin Gamma has a nearly vanishing second derivative with respect to at = 0 or for these values of . Such a thing can be avoided by replacing the sin in the exponential at the ....

....no peaking of the sequence C in the transition region, as occurred in the Bessel coefficients case. The sequence C has a nice compact burstlike behavior. This is an attractive feature because the efficiency will be higher than for the Bessel coefficients. Using the method of stationary phase [9], the integral in Eq. 27 when j j z= can J. Acoustical Soc. of Am. Ronald M. Aarts and A.J.E.M. Janssen 12 be approximated. The stationary points follow from zOE 0 ( 29) and using Eq. 28 we obtain = 2 (1= Gamma =z) 2: 30) Now we get from the stationary phase method C (z) ....

R. Wong. Asymptotic Approximations of Integrals. Academic Press, 1989. J. Acoustical Soc. of Am. Ronald M. Aarts and A.J.E.M. Janssen 20

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Roderick Wong. Asymptotic Approximations of Integrals. Academic Press, 1989.

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Wong, R. (1989) Asymptotic Approximations of Integrals, Academic Press, San Diego, California. 32

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R. Wong, Asymptotic Approximations of Integrals, SIAM, Philadelphia, 2001. 29

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R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, 1989. 16

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R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, 1989.

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R. Wong, Asymptotic approximations of integrals, Academic Press, Inc., Boston, 1989. Academia Sinica Taipei 11529 Taiwan e-mail: hkhwang@stat.sinica.edu.tw 12

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R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, 1989.

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R. Wong, Asymptotic approximations of integrals, Academic Press, Inc., Boston, 1989. Academia Sinica Taipei 11529 Taiwan e-mail: hkhwang@stat.sinica.edu.tw 12

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R. Wong, Asymptotic Approximations of Integrals, Academic Press, New York, 1989.

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