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## Non-Uniform Random Variate Generation (1986)

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Citations: | 997 - 26 self |

### Citations

5031 |
Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images
- Geman, Geman
- 1984
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Citation Context ... f(Y )/f(X), then X ← Y . If g has support over all ofsd , then the Metropolis random walk converges. 6.7. The Gibbs sampler. Among Markov chains that do not use rejection, we cite the Gibbs sampler (=-=Geman and Geman, 1984-=-) which uses the following method to generate Xn+1 given Xn = (Xn,1, . . . , Xn,d): first generate Then generate and continue on until Xn+1,1 ∼ f(·|Xn,2, . . . , Xn,d). Xn+1,2 ∼ f(·|Xn+1,1, Xn,3 . . .... |

3533 | Equations of state calculations by fast computational machine - Metropolis - 1953 |

2033 |
Monte Carlo sampling methods using Markov chains and their applications
- Hastings
- 1970
(Show Context)
Citation Context ...fficiently general construction of stopping times is still lacking. 6.1. The Metropolis-Hastings chain. The Metropolis-Hastings chain (Metropolis, Rosenbluth, Rosenbluth, Teller and Teller, 1953, and =-=Hastings, 1970-=-) can be used for the generation of a random variate with an arbitrary density, provided that some care is taken in its choice of Markov transition probabilities. It requires transition probabilities ... |

1336 |
A method for the construction of minimum-redundancy codes
- Huffman
- 1952
(Show Context)
Citation Context ...ch by comparing U with appropriately picked thresholds. If the cost of setting up this tree is warranted, then one could always permute the leaves to make this into a Huffman tree for the weights pn (=-=Huffman, 1952-=-), which insures that the expected time to find a leaf is not more than one plus the binary entropy, � pn log2(1/pn). n In any case, this value does not exceed log 2 N, where N is the number of possib... |

1249 | An Introduction to Copulas - Nelsen - 1999 |

1208 | Sampling-based approaches to calculating marginal densities - Gelfand, Smith - 1990 |

537 | Reversible Markov chains and random walks on graphs. - Aldous, Fill - 2010 |

536 | Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Structures Algorithms 9 - Propp, Wilson - 1996 |

294 |
A note on the generation of random normal deviates
- Box, Muller
- 1958
(Show Context)
Citation Context ... integer, or a random integer with a small mean. It is remarkable that one can obtain the normal and indeed all stable distributions using simple transformations with k = 2. In the Box-Müller method (=-=Box and Müller, 1958-=-), a pair of independent standard normal random variates is obtained by setting �� (X, Y ) = log(1/U1) cos(2πU2), � � log(1/U1) sin(2πU2) , where U1, U2 are independent uniform [0, 1] random variates.... |

273 |
Probability metrics and the stability of stochastic models,
- Rachev
- 1991
(Show Context)
Citation Context ...zed. For example, the Wasserstein metric d2(F, G) is the minimal value over all couplings of (X, Y ) of �s{(X − Y ) 2 }. That minimal coupling occurs when (X, Y ) = (F inv (U), G inv (U)) (see, e.g., =-=Rachev, 1991-=-). Finally, if we wish to simulate the maximum M of n i.i.d. random variables, each distributed as X = F inv (U), then noting that the maximum of n i.i.d. uniform [0, 1] random variables is distribute... |

248 |
The Classical Moment Problem
- Akhiezer
- 1965
(Show Context)
Citation Context ...less additional information is available. For one thing, there may not be a unique solution. A sufficient condition that guarantees the uniqueness is Carleman’s condition ∞� n=0 1 − |µ2n| 2n = ∞ (see =-=Akhiezer, 1965-=-). Sufficient conditions in terms of a density f exist, such as Krein’s condition � − log(f(x)) 1 + x2 dx = ∞, combined with Lin’s condition, applicable to symmetric and differentiable densities, whic... |

239 | A Method for Simulating Stable Random Variables - Chambers, Mallows, et al. - 1976 |

211 | On the Statistical Analysis of Dirty Pictures with Discussion: - Besag - 1986 |

208 | A Course in Density Estimation
- Devroye
- 1987
(Show Context)
Citation Context ...ers not shown here included the Pólya-Eggenberger distribution (see Johnson and Kotz, 1969) and the hyper-Poisson distribution. For all discrete log-concave distributions having a mode at m, we have (=-=Devroye, 1987-=-a): pm+k ≤ pm min � 1, e 1−pm|k|� , all k. Mimicking the development for densities, we may use rejection from a curve that is flat in the middle, and has two exponential tails. After taking care of ro... |

154 |
Spatial statistics and Bayesian computation
- Besag, Green
- 1993
(Show Context)
Citation Context ...d . Repeat forever: Generate U uniformly on [0, 1]. Set Y ← Uf(X). Repeat Generate X uniformly on [0, 1] d Until f(X) ≥ Y . This algorithm is also known as the slice sampler (Swendsen and Wang, 1987; =-=Besag and Green, 1993-=-). In the first step of each iteration, given X, Y is produced with the conditional distribution required, that is, the uniform distribution on [0, f(X)]. Given Y , a new X is generated uniformly on t... |

145 | Metropolized Independent Sampling with Comparisons to Rejection Sampling and Importance Sampling
- Liu
- 1996
(Show Context)
Citation Context ...roduces an i.i.d. sequence of random variates with density f. Otherwise, if q > 0 there is convergence (in total variation, see below), and even geometric convergence at the rate 1 − inf x q(x)/f(x) (=-=Liu, 1996-=-). One thus should try and match q as well as possible to f. 6.3. The discrete Metropolis chain. The chains given above all remain valid if the state space is finite, provided that the density f(x) an... |

137 | Finite Markov chains and algorithmic applications - Häggström - 2002 |

119 | Simulation of non-homogenous poisson process by thinning - Lewis, Shedler - 1979 |

116 | Bayesian image analysis: an application to single photon emission tomography, - Geman, McClure - 1985 |

99 | Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension - Caflisch, Morokoff, et al. - 1997 |

99 | The complexity of nonuniform random number generation - Knuth, Yao - 1976 |

95 | Approximating integrals via Monte Carlo and deterministic methods - Swartz, Evans - 2000 |

90 | An interruptible algorithm for perfect sampling via Markov chains. Annals of Applied Probability - Fill - 1998 |

80 | Geometric ergodicity of Metropolis algorithms. Stochastic Process
- Jarner, Hansen
- 2000
(Show Context)
Citation Context ...f the chains, started at X0, we know that Xn has a density for n > 0.) Under additional restrictions on q, e.g., q(x, y) inf inf > 0, x y f(x) we know that Vn ≤ exp(−ρn) for some ρ > 0 (Holden, 1998; =-=Jarner and Hansen, 1998-=-). For more on total variation convergence of Markov chains, see Meyn and Tweedie (1993). 6.6. The Metropolis random walk. The Metropolis random walk is the Metropolis chain obtained by setting q(x, y... |

71 |
On the Frequency of Numbers containing Prime Factors of a Certain Relative Magnitude, Arkiv för Matematik, Astronomi och Fysik 22 A
- Dickman
- 1930
(Show Context)
Citation Context ... for example, when W = U 1/α , where U is uniform [0, 1] and α > 0 leads to the characteristic function ϕ(t) = e α � 1 e 0 itx −1 x dx . For the particular case α = 1, we have Dickman’s distribution (=-=Dickman, 1930-=-), which is the limit law of n� 1 iZi n i=1 where Zi is Bernoulli (1/i). None of the three representations above (infinite series, Fourier transform, limit of a discrete sum) gives a satisfactory path... |

71 | Automatic Nonuniform Random Variate Generation - Hormann, Leydold, et al. - 2004 |

70 | The contraction method for recursive algorithms
- Rösler, Rüschendorf
- 2001
(Show Context)
Citation Context ... in any case, it is not more than 4 + � 24s{h(0)X}. 21 j=1s4.7. A distributional identity. The theory of fixed points and contractions can be used to derive many limit laws in probability (see, e.g., =-=Rösler and Rüschendorf, 2001-=-). These often are described as distributional identities known to have a unique solution. For example, the identity X L = W (X + 1) where W is a fixed random variable on [0, 1] and X ≥ 0 sometimes ha... |

65 | Computer methods for sampling from gamma, beta, Poisson, and binomial distributions. Computing 12 - Ahrens, Dieter |

60 | Stationarity detection in the initial transient problem - Asmussen, Glynn, et al. - 1992 |

48 | Perpetuities with thin tails
- Goldie, Grübel
- 1996
(Show Context)
Citation Context ...X. By iterating the identity, we see that if the solution exists, it can be represented as X L = W1 + W1W2 + W1W2W3 + · · · , where the Wi’s are i.i.d. These are known as perpetuities (Vervaat, 1979, =-=Goldie and Grübel, 1996-=-). Some more work in the complex domain can help out: for example, when W = U 1/α , where U is uniform [0, 1] and α > 0 leads to the characteristic function ϕ(t) = e α � 1 e 0 itx −1 x dx . For the pa... |

48 | Exact sampling for Bayesian inference: towards general purpose algorithms - Green, Murdoch - 1999 |

44 |
On the height of trees
- RENYI, SZEKERES
- 1967
(Show Context)
Citation Context ... − jt) j+1 This is sufficient to apply the alternating series method (Devroye, 1991). 20s4.5. An infinite series. Some densities are known as infinite series. Examples include the theta distribution (=-=Rényi and Szekeres, 1967-=-) with distribution function ∞� F (x) = (1 − 2j 2 x 2 )e −j2x 2 ∞� j 2 e −π2j 2 /x 2 , x > 0, j=−∞ = 4π5/2 x 3 and the Kolmogorov-Smirnov distribution (Feller, 1948) with distribution function ∞� F (x... |

39 | Stable densities under change of scale and total variation inequalities - Kanter - 1975 |

38 | Strategies for Quasi-Monte Carlo - Fox - 1999 |

35 |
On the Kolmogorov-Smirnov limit theorems for empirical distributions
- FELLER
- 1948
(Show Context)
Citation Context ...the theta distribution (Rényi and Szekeres, 1967) with distribution function ∞� F (x) = (1 − 2j 2 x 2 )e −j2x 2 ∞� j 2 e −π2j 2 /x 2 , x > 0, j=−∞ = 4π5/2 x 3 and the Kolmogorov-Smirnov distribution (=-=Feller, 1948-=-) with distribution function ∞� F (x) = 1 − 2 (−1) j e −2j2x 2 , x > 0. j=1 In the examples above, it is relatively straightforward to find tight bounding curves, and to apply the alternating series m... |

34 |
The Theory of Approximation
- Jackson
- 1930
(Show Context)
Citation Context ...unction of all moments up to the k-th. The series truncated at index n has an error not exceeding � (r+1) |f | Cr (1 − x2 ) 1/4nr−1/2 where r is an integer and Cr is a constant depending upon r only (=-=Jackson, 1930-=-). Rejection with dominating curve of the form C ′ /(1 − x 2 ) 1/4 (a symmetric beta) and with the alternating series method can thus lead to a generator (Devroye, 1989). Let The situation is much mor... |

34 | A fast, easily implemented method for sampling from decreasing or symmetric unimodal density functions - Marsaglia, Tsang - 1984 |

33 | A rejection technique for sampling from T-concave distributions - Hörmann - 1995 |

33 | Binomial random variate generation - KACHITVICHYANUKUL, B - 1988 |

32 | On simulation from infinitely divisible distributions - Bondesson - 1982 |

32 | Adaptive rejection Metropolis sampling - Gilks, Best, et al. - 1995 |

31 | A fast procedure for generating normal random variables - Marsaglia, Maclaren, et al. - 1964 |

30 | Generating gamma variates by a modified rejection technique - Ahrens, Dieter - 1982 |

30 | AIDS—The Problem in
- Freedman
- 1987
(Show Context)
Citation Context ...ty f exist, such as Krein’s condition � − log(f(x)) 1 + x2 dx = ∞, combined with Lin’s condition, applicable to symmetric and differentiable densities, which states that x|f ′ (x)|/f(x) ↑ ∞ as x → ∞ (=-=Lin, 1997-=-, Krein, 1944, Stoyanov, 2000). Whenever the distribution is of compact support, the moments determine the distribution. In fact, there are 19svarious ways for reconstructing the density from the mome... |

30 | A simple method for generating gamma variables - Marsaglia, Tsang - 2001 |

29 | Automatic sampling with the ratio-of-uniforms method - Leydold - 2000 |

28 | A one-table method for sampling from continuous and discrete distributions - Ahrens - 1995 |

27 |
On generating random variates from an empirical distribution
- Chen, Asau
- 1974
(Show Context)
Citation Context ...binary search, 6swhich would take O(log 2 N) time. As mentioned above, the Huffman tree is optimal, and thus better, but requires more work to set up. 1.5. Guide tables. Hash tables, or guide tables (=-=Chen and Asau, 1974-=-), have been used to accelerate the inversion even further. Given the size of the universe, N, we create a hash table with N entries, and store in the i-th entry the value of X if U were i/N, 0 ≤ i < ... |

27 | Perfect Slice Samplers - Mira, Moller, et al. |

26 | Random variable generation using concavity properties of transformed densities - Evans, Swartz - 1998 |

25 |
A simple algorithm for generating random variates with a log-concave density
- Devroye
- 1984
(Show Context)
Citation Context ...ibution (with density proportional to 1/(e x + e −x + a), a > −2), the hyperbolic secant distribution (with distribution function (2/π) arctan(e x )), and Kummer’s distribution (for a definition, see =-=Devroye, 1984-=-a). Also, we should consider simple nonlinear transformations of other random variables. Examples include arctan X with X Pearson IV, log |X| (X being Student t for all parameters), log X (for X gamma... |

24 | The generation of gamma variables with non-integral shape parameter - Cheng - 1977 |

20 |
A simple generator for discrete log-concave distributions
- Devroye
- 1987
(Show Context)
Citation Context ...ers not shown here included the Pólya-Eggenberger distribution (see Johnson and Kotz, 1969) and the hyper-Poisson distribution. For all discrete log-concave distributions having a mode at m, we have (=-=Devroye, 1987-=-a): pm+k ≤ pm min � 1, e 1−pm|k|� , all k. Mimicking the development for densities, we may use rejection from a curve that is flat in the middle, and has two exponential tails. After taking care of ro... |

18 | Sampling from general distributions by suboptimal division of domains - Ahrens - 1993 |

18 | A note on Linnik's distribution - Devroye - 1990 |

17 | A Bernoulli factory - Keane, O'Brien - 1994 |

14 | Letter to the editors - Best - 1978 |

14 | Exact sampling for Bayesian inference: Unbounded state spaces - Murdoch, J - 2000 |

12 | Some simple gamma variate generators - Cheng, Feast - 1979 |

12 | The squeeze method for generating gamma variates - Marsaglia - 1977 |

11 | Sampling from binomial and Poisson distributions : A method with bounded computation times - AHRENS, U - 1980 |

11 |
On the computer generation of random variables with a given characteristic function
- Devroye
- 1981
(Show Context)
Citation Context ...distribution function ∞� F (x) = 1 − 2 (−1) j e −2j2x 2 , x > 0. j=1 In the examples above, it is relatively straightforward to find tight bounding curves, and to apply the alternating series method (=-=Devroye, 1981-=-a and 1997). 4.6. Hazard rates. Let X be a positive random variable with density f and distribution function F . Then the hazard rate, the probability of instantaneous death given that one is still al... |

11 | Perfect simulation from the quicksort limit distribution - Devroye, Fill, et al. - 2000 |

10 | Gamma variate generators with increased shape parameter range - Cheng, Feast - 1980 |

10 | On random variate generation when only moments or Fourier coefficients are known
- DEVROYE
- 1989
(Show Context)
Citation Context ...ndex up to n, then the error made is not more than ∞� Rn+1 = k=n+1 � a 2 k + b 2 k . If we know bounds on Rn, then one can use rejection with an alternating series method to generate random variates (=-=Devroye, 1989-=-). A particularly simple situation occurs when only the cosine coefficients are nonzero, as occurs for symmetric distributions. Assume furthermore that the Fourier cosine series coefficients ak are co... |

10 |
The complexity of generating an exponentially distributed variate
- Flajolet, Saheb
- 1986
(Show Context)
Citation Context ...2sabout the individual bits in the binary expansions of the pn’s, so that we will, even for discrete distributions, consider only the ram model. Noteworthy is that attempts have been made (see, e.g., =-=Flajolet and Saheb, 1986-=-) to extend the pure bit model to obtain approximate algorithms for random variables with densities. 1.1. The inversion method. For a univariate random variable, the inversion method is theoretically ... |

10 | The transformed rejection method for generating Poisson random variables - Hörmann - 1993 |

9 | A simple algorithm for the computer generation of random samples from a Student's t or symmetric beta distribution - Best - 1978 |

9 | The Series Method in Random Variate Generation and Its Application to the Kolmogorov-Smimov Distribution
- Devroye
- 1981
(Show Context)
Citation Context ...distribution function ∞� F (x) = 1 − 2 (−1) j e −2j2x 2 , x > 0. j=1 In the examples above, it is relatively straightforward to find tight bounding curves, and to apply the alternating series method (=-=Devroye, 1981-=-a and 1997). 4.6. Hazard rates. Let X be a positive random variable with density f and distribution function F . Then the hazard rate, the probability of instantaneous death given that one is still al... |

9 | Density approximation and exact simulation of random variables that are solutions of fixed-point equations - Devroye, Neininger - 2002 |

9 | A universal generator for discrete log-concave distributions - Hörmann - 1994 |

8 | An alias method for sampling from the normal distribution - Ahrens, Dieter - 1989 |

8 | A convenient sampling method with bounded computation times for Poisson distributions,” in: The Frontiers of Statistical Computation, Simulation and Modeling, (edited by - Ahrens, Dieter - 1991 |

8 | Random variate generation for unimodal and monotone densities
- DEVROYE
- 1984
(Show Context)
Citation Context ...ibution (with density proportional to 1/(e x + e −x + a), a > −2), the hyperbolic secant distribution (with distribution function (2/π) arctan(e x )), and Kummer’s distribution (for a definition, see =-=Devroye, 1984-=-a). Also, we should consider simple nonlinear transformations of other random variables. Examples include arctan X with X Pearson IV, log |X| (X being Student t for all parameters), log X (for X gamma... |

8 |
Algorithms for generating discrete random variables with a given generating function or a given moment sequence
- DEVROYE
- 1991
(Show Context)
Citation Context ...mple, if we take t > 0 arbitrary, then we can define �j i=0 pnj = (−1)j−i� � j k(it) i j!tj and note that 0 ≤ pnj − pj ≤ 1 − 1. (1 − jt) j+1 This is sufficient to apply the alternating series method (=-=Devroye, 1991-=-). 20s4.5. An infinite series. Some densities are known as infinite series. Examples include the theta distribution (Rényi and Szekeres, 1967) with distribution function ∞� F (x) = (1 − 2j 2 x 2 )e −j... |

8 | Geometric convergence of the Metropolis–Hastings simulation algorithm
- HOLDEN
- 1998
(Show Context)
Citation Context ...ous versions of the chains, started at X0, we know that Xn has a density for n > 0.) Under additional restrictions on q, e.g., q(x, y) inf inf > 0, x y f(x) we know that Vn ≤ exp(−ρn) for some ρ > 0 (=-=Holden, 1998-=-; Jarner and Hansen, 1998). For more on total variation convergence of Markov chains, see Meyn and Tweedie (1993). 6.6. The Metropolis random walk. The Metropolis random walk is the Metropolis chain o... |

8 | A mixture representation of the Linnik distribution - Kotz, Ostrovskii - 1996 |

7 | Methods for generating random variates with Polya characteristic functions
- Devroye
- 1984
(Show Context)
Citation Context ...ibution (with density proportional to 1/(e x + e −x + a), a > −2), the hyperbolic secant distribution (with distribution function (2/π) arctan(e x )), and Kummer’s distribution (for a definition, see =-=Devroye, 1984-=-a). Also, we should consider simple nonlinear transformations of other random variables. Examples include arctan X with X Pearson IV, log |X| (X being Student t for all parameters), log X (for X gamma... |

6 | Algorithm 599. Sampling from gamma and Poisson distributions - AHRENS, KOHRT, et al. - 1983 |

6 | A note on gamma variate generators with shape parameter less than unity - Best - 1983 |

6 |
An automatic method for generating random variables with a given characteristic function
- Devroye
- 1988
(Show Context)
Citation Context ...en hazard rate, then for a dhr (decreasing hazard rate) distribution, we can use g = h(0). In that case, the expected number of iterations before halting iss{h(0)X}. However, we can dynamically thin (=-=Devroye, 1986-=-c) by lowering g as values h(Yi) trickle in. In that case, the expected time is finite when X has a finite logarithmic moment, and in any case, it is not more than 4 + � 24s{h(0)X}. 21 j=1s4.7. A dist... |

6 | A simple universal generator for continuous and discrete univariate T-concave distributions - Leydold - 2001 |

5 | Polar generation of random variates with the t distribution - Bailey - 1994 |

5 | A note on transformed density rejection - Leydold - 2000 |

4 |
On a class of unimodal distributions
- Laha
(Show Context)
Citation Context ...es, U is a uniform [0, 1] random variable, and α ∈ (0, 1], then Y/Z with Z = (E1 + E2 U<α) 1 α is symmetric stable of parameter (α), with characteristic function e−|t|α. The Linnik-Laha distribution (=-=Laha, 1961-=-) with parameter α ∈ (0, 1] is described by Here Y/Z with has the desired distribution. Z = ϕ(t) = 1 . 1 + |t| α � α + 1 − � (α + 1) 2 − 4αU 2U 4.2. Fourier coefficients. Assume that the density f, su... |

3 |
The computer generation of Poisson random variables
- Devroye
- 1981
(Show Context)
Citation Context ...distribution function ∞� F (x) = 1 − 2 (−1) j e −2j2x 2 , x > 0. j=1 In the examples above, it is relatively straightforward to find tight bounding curves, and to apply the alternating series method (=-=Devroye, 1981-=-a and 1997). 4.6. Hazard rates. Let X be a positive random variable with density f and distribution function F . Then the hazard rate, the probability of instantaneous death given that one is still al... |

3 | Simulating perpetuities,” Methodology and Computing - Devroye - 2001 |

3 | The generation of binomial random variates - Hormann - 1993 |

3 |
On an extrapolation problem of A.N
- Krein
- 1944
(Show Context)
Citation Context ... such as Krein’s condition � − log(f(x)) 1 + x2 dx = ∞, combined with Lin’s condition, applicable to symmetric and differentiable densities, which states that x|f ′ (x)|/f(x) ↑ ∞ as x → ∞ (Lin, 1997, =-=Krein, 1944-=-, Stoyanov, 2000). Whenever the distribution is of compact support, the moments determine the distribution. In fact, there are 19svarious ways for reconstructing the density from the moments. An examp... |

2 | The analysis of some algorithms for generating random variates with a given hazard rate
- Devroye
- 1986
(Show Context)
Citation Context ...en hazard rate, then for a dhr (decreasing hazard rate) distribution, we can use g = h(0). In that case, the expected number of iterations before halting iss{h(0)X}. However, we can dynamically thin (=-=Devroye, 1986-=-c) by lowering g as values h(Yi) trickle in. In that case, the expected time is finite when X has a finite logarithmic moment, and in any case, it is not more than 4 + � 24s{h(0)X}. 21 j=1s4.7. A dist... |

2 | Generating sums in constant average time,” in - Devroye - 1988 |

2 | Random variate generation in one line of code,” in: 1996 Winter Simulation Conference Proceedings, (edited by - Devroye - 1996 |

2 | Fonctions convexes de Polya. Publications de l’Institut de Statistique des Universités de - Dugué, Girault - 1955 |

2 | Adaptive rejection for Gibbs sampling - GILKS, WILD - 1992 |

2 | Minh, “Generating gamma variates - Le - 1988 |

1 | Simulating theta random variates - Devroye - 1997 |

1 | Universal generators for correlation induction,” in: COMPSTAT 94, (edited by - Hörmann, Derflinger - 1994 |

1 | Algorithm 678: BTPEC: sampling from the binomial distribution - Kachitvichyanukul, Schmeiser - 1989 |

1 | On building random variables of a given distribution - Letac - 1975 |

1 |
Linear forms and statistical criteria,I,II,” Selected Translations
- Linnik
- 1962
(Show Context)
Citation Context ...ations and a thorough discussion on these families of distributions. The paper by Devroye (1990) contains other examples with k = 3, including Sα,0E 1 α , which has the so-called Linnik distribution (=-=Linnik, 1962-=-) with characteristic function ϕ(t) = 1 , 0 < α ≤ 2. 1 + |t| α See also Kotz and Ostrovskii (1996). It also shows that Sα,1E 1 α , has the Mittag-Leffler distribution with characteristic function ϕ(t)... |