### Table I. Enumeration data for the first three odd moments of the distribution of endpoint displacements along the x axis, for SAWS on hypercubic lattices, when the walk starts at the origin with an excluded site at x = -1.

1988

### Table 2 Posterior Odds Ratios and Moments for Six Macroeconomic Time Series* P.O.R. in favor of -------r ------ -----d x100----- ------n ------

1992

"... In PAGE 15: ... Of course, the actual data used are annual in each case. Results are reported in Table2 .... In PAGE 15: ... Simple arithmetic shows that except for error due to numerical approximation, the Next s odds ratio should be the ratio of the r = 1 odds ratio for that row to the r = 1 odds ratio for the next row, a relationship that is borne out up to the number of places accuracy that numerical standard errors would indicate. These indicators of numerical accuracy are not reported here, but they are used to choose the number of digits reported in Table2 just as they were in Table 1. As the prior parameter s increases the posterior mean of r increases and its posterior standard deviation decreases monotonically (within the limits of numerical accuracy) in every case.... In PAGE 16: ...For these series, the evidence against the unit root hypothesis is strongest when s = 9 or s = 29. Comparison of results for different time series in Table2 is generally consistent with the findings of other investigators for these data: e.g.... In PAGE 16: ... In view of the placement of the Schotman-van Dijk prior and the prior density 10r 9 in the case s = 9, this correspondence is quite reasonable. Any differences between Schotman and van Dijk [31] and Table2 seem plausibly attributable to the richer model specification and different priors used here. Phillips [25] uses a variant of the model (4) and obtains posterior odds ratios in favor of a unit root: real GNP, .... In PAGE 16: ...apita GNP, .25; unemployment, .015; consumer prices, 7.7; velocity, 2.0. These findings are substantially different from any of those reported in Table2 , and from the results of Schotman and van Dijk, indicating (as one would expect) the sensitivity of the posterior odds ratio to the specificatio of the model and prior distributions. Posterior moments for r and d are consistent with those reported by Schotman and van Dijk [31], and with parameter estimates for more distantly related models taken up by other investigators.... In PAGE 17: ...ensity attains its maximum ratio of almost 2;1 at around n = 5.. These results are consistent with the strong evidence for leptokurtosis reported in Geweke ([11]. Corresponding posterior densities for the other five time series are qualitatively similar, but with changes in location and scale suggested by the posterior means and standard deviations presented in Table2 . These densities are displayed in Geweke [10].... In PAGE 24: ... Table2 (continued) P.... ..."

Cited by 4

### Table 2. Enumeration data for the first three odd moments of the distribution of endpoint displacements along the x axis, for SAWS on the square, simple cubic and hypercubic lattices, when the half-line from -m to the origin is excluded, and the walker starts from the origin.

1988

"... In PAGE 6: ...D Considine and S Redner Table2 . (continued) Four dimensions N 3 4 5 6 7 8 9 10 11 12 I3 (XN ) 0.... ..."

### Table 1: Moments of the transformation Z = jXj Px = 1 Moments

"... In PAGE 3: ... It is a well known fact that the trans- formed random variable Y is non-Gaussian and its odd- order moments are non-zero. A Monte Carlo (MC) simulation with 25 million samples has been performed and the resulting moments are shown in Table1 along with the estimates of the Higher Order Unscented Fil- ter and the standard Unscented Filter (UF). The pa- rameters for the HOUF are derived from equation 22 and listed in Table 2.... ..."

### Table 5: Properties of BME pdf apos;s when one of p and q is an integer and the other is an odd multiple of 1/2. that its moments are n!=(2n + 1) and recalling that the power series for the arctan function has coe cients (2n + 1)?1, see 2.5.9 on p. 53 of Wilf [35], we see that

1999

"... In PAGE 23: ...5) We note that the cases v(1=2; 1=2; t) (t) and v(1; 1; t) E1(t) were used to study the Laguerre-series algorithm in Abate, Choudhury and Whitt [10]; see Tables 1 and 2 there. Table5 contains cases in which one of p and q is integer while the other is an odd multiple of 1/2. The rst entry v(1=2; 1; t) is a curious pdf.... In PAGE 24: ...Table5 is determined from an integral representation of the Bessel function K0(t); see 9.... ..."

Cited by 6

### Table 2: First moments of the mode- uctuation distribution P (W ) for the odd and even case. By de nition Nosc(E) uctuates around zero, see gure 3, and furthermore it can be shown [19] that the second moment of Nosc(E) tends asymptotically to the saturation value 1(E) of the rigidity. In [5, 6] the conjecture has been put forward that classically chaotic systems should display a universal Gaussian behavior in the limit E ! 1 PGauss(W ) = 1 p2 e?1 2W2 ; (15) whereas classically integrable systems should display non{Gaussian distributions P (W ).

### Table 2. (continued)

1988

"... In PAGE 5: ...1625 Table2 . Enumeration data for the first three odd moments of the distribution of endpoint displacements along the x axis, for SAWS on the square, simple cubic and hypercubic lattices, when the half-line from -m to the origin is excluded, and the walker starts from the origin.... ..."

### Table 3. The coe cients Dp n for the rst 6 moments

"... In PAGE 6: ... But we can also express mp in another form using the C coe cients: mp = p X n=1 Bp nSn+1 1 = p X n=1 Bp n(S2 1 + n X k=2 Sk+1 2 ) = p X n=1 Bp nS2 1 + p X n=1 Bp n n X k=2 Sk+1 2 = Cp 1;1S2 1 + p X n=2 Cp n;1Sn+1 2 = Cp 1;1S2 1 + Cp 2;1S3 2 + p X n=3 Cp n;2Sn+1 2 = Cp 1;1S2 1 + Cp 2;1S3 2 + Cp 3;2S4 2 + Cp 4;2S5 3 + + p X n=k Cp n;k?1Sn+1 k = p X n=1 Cp n;dn2 eSn+1 dn+12 e; valid for p 1. With C0 0;0 = 1 and Cp 0;0 = 0 for p 1, it is again possible to use the general form for p 0 mp = p X n=0 Cp n;dn2 eSn+1 dn+12 e: We can then prove that Cp n;dn2 e has the follow- ing properties: p! Cp n;dn2 e 0 Cp 1;1 = p X k=1 Bp k = 1 Cp p;dp2e = Bp p = p! Cp n;dn2 e = ( 0 2Cp n?1;dn2 e p = 1; 3; 5; 7; p = 2; 4; 6; 8; ; for n = 2; 4; 6; 8; : The coe cients Cp n;dn2 e for the rst 6 moments are shown in Table3 . Comparing the above coef- cients with the previous two kinds of coe cients we can see that now all coe cients Cp n;dn2 e, are non-negative and equal to or less than p!, and al- most half of the coe cients for odd valued p are zero.... ..."

### Table 4.1: Sobolev smoothness for minimal support scaling functions with N vanishing moments However, one can go much further. Eirola [8] used the notion of TR as a positive operator mapping a cone K to itself to develop asymptotic estimates for the spectral radius of TR as N ! 1. For m odd, de ne the cone KJ = 8 lt; :u 2 K0 \ EJ : u(!) = J X j=0 uj(1 ? cos !)j ; uj 09 = ; : Eirola de nes an analogous cone for m even. In [14] we prove

### Table 7: Statistical Moments, Traded and Non-Traded Sectors Variable Israel Data Model

"... In PAGE 7: ... The first method, which is marked in the Table as A is taken from Helpman and Drazen (1987) where the ratio is extracted from a general price function which is consistent with the model definitions. The second method, which is marked in Table7 as B is taken from Razin and Cuckerman (1976) where the price ratio is extracted based on CPI, according to the consumption apos;s classification. The third method which is marked in Table 7 as C is taken from Meridor and Pesach (1994) where the ratio is calculated according to the ratio of import prices to output prices, C1, and the ratio of export prices to output prices, C2.... In PAGE 7: ... The second method, which is marked in Table 7 as B is taken from Razin and Cuckerman (1976) where the price ratio is extracted based on CPI, according to the consumption apos;s classification. The third method which is marked in Table7 as C is taken from Meridor and Pesach (1994) where the ratio is calculated according to the ratio of import prices to output prices, C1, and the ratio of export prices to output prices, C2. The data availability for methods B and C are from 1981.... In PAGE 22: ...25 It turns out that this case improves the closeness of the second moments predicted by the model to the data second moments. In Table7 we extend the analysis to the dynamics of each sector separately as well as the dynamic of real exchange rate extracted from the price ratio between traded and non-traded goods. We compare the moments of these variables to the moment from the Israeli data, using the parameters of case 12 in Table 6 (Appendix C) as a benchmark.... In PAGE 24: ...phenomenon exists both in the data and in the model ( Table7 ). This result might be at odds with intuition based on a one sector model version of a small open economy and hence can be viewed as surprising.... ..."