Abstract

The distribution of the sum of log-normal variates is shown for most cases of interest to be very accurately represented by a log-normal distribution instead of a normal or Rayleigh distribution that might be expected from the central-limit theorem. As a result, observation of the log-normal distribution for the fluctuations of flux received after propagation through a random medium can be readily explained, regardless of the size of the receiving aperture.

In every case, the log-normal distribution is a better representation than the normal for the distribution of the sum of log-normal variates. However, in some cases not even the log-normal distribution is very accurate. The questions of convergence and accuracy are examined in detail.

© 1968 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Mediun, (McGraw-Hill Book Company, New York, 1961), p. 208.
  2. See Ref. 1, p. 229.
  3. D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
    [CrossRef]
  4. S. K. Grace and S. N. Miller, Proc. IEEE 54, 1593 (1966).
    [CrossRef]
  5. P. Beckmann, J. Res. Natl. Bur. Std. (U. S.) D68, 723 (1964).
  6. G. R. Heidbreder and R. L. Mitchell, IEEE Trans. Aerospace Electron Systems 3, 5 (1967).
    [CrossRef]
  7. J. Aitchison and J. A. C. Brown, The Log-Normal Distribution (Cambridge University Press, London, 1957).
  8. H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), p. 185.
  9. See Ref. 8, p. 131.
  10. See Ref. 8, p. 221.
  11. D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 10 (1967).

1967 (3)

D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
[CrossRef]

G. R. Heidbreder and R. L. Mitchell, IEEE Trans. Aerospace Electron Systems 3, 5 (1967).
[CrossRef]

D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 10 (1967).

1966 (1)

S. K. Grace and S. N. Miller, Proc. IEEE 54, 1593 (1966).
[CrossRef]

1964 (1)

P. Beckmann, J. Res. Natl. Bur. Std. (U. S.) D68, 723 (1964).

Aitchison, J.

J. Aitchison and J. A. C. Brown, The Log-Normal Distribution (Cambridge University Press, London, 1957).

Beckmann, P.

P. Beckmann, J. Res. Natl. Bur. Std. (U. S.) D68, 723 (1964).

Brown, J. A. C.

J. Aitchison and J. A. C. Brown, The Log-Normal Distribution (Cambridge University Press, London, 1957).

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), p. 185.

Fried, D. L.

D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 10 (1967).

D. L. Fried, G. E. Mevers, and M. P. Keister, J. Opt. Soc. Am. 57, 787 (1967).
[CrossRef]

Grace, S. K.

S. K. Grace and S. N. Miller, Proc. IEEE 54, 1593 (1966).
[CrossRef]

Heidbreder, G. R.

G. R. Heidbreder and R. L. Mitchell, IEEE Trans. Aerospace Electron Systems 3, 5 (1967).
[CrossRef]

Keister, M. P.

Mevers, G. E.

Miller, S. N.

S. K. Grace and S. N. Miller, Proc. IEEE 54, 1593 (1966).
[CrossRef]

Mitchell, R. L.

G. R. Heidbreder and R. L. Mitchell, IEEE Trans. Aerospace Electron Systems 3, 5 (1967).
[CrossRef]

Schmeltzer, R. A.

D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 10 (1967).

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Mediun, (McGraw-Hill Book Company, New York, 1961), p. 208.

Appl. Opt. (1)

D. L. Fried and R. A. Schmeltzer, Appl. Opt. 6, 10 (1967).

IEEE Trans. Aerospace Electron Systems (1)

G. R. Heidbreder and R. L. Mitchell, IEEE Trans. Aerospace Electron Systems 3, 5 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Res. Natl. Bur. Std. (U. S.) (1)

P. Beckmann, J. Res. Natl. Bur. Std. (U. S.) D68, 723 (1964).

Proc. IEEE (1)

S. K. Grace and S. N. Miller, Proc. IEEE 54, 1593 (1966).
[CrossRef]

Other (6)

V. I. Tatarski, Wave Propagation in a Turbulent Mediun, (McGraw-Hill Book Company, New York, 1961), p. 208.

See Ref. 1, p. 229.

J. Aitchison and J. A. C. Brown, The Log-Normal Distribution (Cambridge University Press, London, 1957).

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), p. 185.

See Ref. 8, p. 131.

See Ref. 8, p. 221.

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Figures (3)

Fig. 1
Fig. 1

Regions of convergence where sum of n log-normal variates is approximately log normal. A, convergence for both log-normal and normal approximations; B, convergence for log-normal approximation; C, convergence uncertain.

Fig. 2
Fig. 2

Cumulative-probability distribution for sum of n log-normal variates with q(1)=2. The curves represent the log-normal approximation to the distribution function with one correction term. A true log-normal plots as a straight line.

Fig. 3
Fig. 3

Cumulative-probability distribution for log-normal plotted on normal-probability coordinates. The abscissa is (x−〈x〉)/σx, where 〈x〉 and σx are the mean and standard deviation of the log-normal distribution function. A true normal plots as a straight line.

Tables (1)

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Table I Lower-order coefficients for central moments.

Equations (37)

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d Λ ( x ) = [ ( 2 π ) 1 2 σ x ] - 1 exp [ - 1 2 ( ln x - λ σ ) 2 ] d x .
Λ ( x ) = N [ y = ( ln x - λ ) / σ ] .
ρ = x / x 0 = exp ( σ 2 / 2 ) .
μ 2 = ( x - x ) 2 = q 2 α 1 2 ,
q 2 = ρ 2 - 1 = exp ( σ 2 ) - 1.
α n = x n = ρ n ( n - 1 ) α 1 n .
μ n = q n α 1 n k = 0 n ( n - 2 ) C k q k .
μ n μ 2 n / 2 = k = 0 n ( n - 2 ) C k q k .
γ 1 = μ 3 / μ 2 3 2 = 3 q + q 3
γ 2 = μ 4 μ 2 2 - 3 = 16 q 2 + 15 q 4 + 6 q 6 + q 8 .
ln ϕ ( t ) = k = 0 χ k ( i t ) k k ! ,
ϕ ( t ) = 0 e i t x d Λ ( x ) .
χ k = α 1 k q k [ k k - 2 q k - 2 + + q k ( k - 2 ) ] .
γ k = χ k + 2 / μ 2 k / 2 + 1 = ( k + 2 ) k q k + + q k ( k + 2 ) ,
F ( x ) = F 1 ( ξ - x ) d F 2 ( ξ ) ,
α 1 = α 1 ( 1 ) + α 1 ( 2 ) + + α 1 ( n ) μ 2 = μ 2 ( 1 ) + μ 2 ( 2 ) + + μ 2 ( n ) μ 3 = μ 3 ( 1 ) + μ 3 ( 2 ) + + μ 3 ( n ) } .
α 1 = n α 1 ( 1 ) μ 2 = n μ 2 ( 1 ) μ 3 = n μ 3 ( 1 ) } .
γ k = n - k / 2 γ k ( 1 ) = n - k / 2 [ ( k + 2 ) k q k + + q k ( k + 2 ) ] .
f ( x ) = b 0 p 0 ( x ) f * ( x ) + b 1 p 1 ( x ) f * ( x ) + ,
p m ( x ) p n ( x ) f * ( x ) d x = { 1 m = n 0 m n .
p n ( x ) = K | α 0 * α 1 * α n * α 1 * α n - 1 * α n * α 2 n - 1 * 1 x x n | ,
b n = p n ( x ) f ( x ) d x .
q * = μ 2 1 2 α 1 = [ n μ 2 ( 1 ) ] 1 2 n α 1 ( 1 ) = n - q ( 1 ) ,
μ n * ( μ 2 * ) n / 2 = k = 0 n ( n - 2 ) C k n - k / 2 q k ( 1 ) ,
γ k * = ( k + 2 ) k n - k / 2 q k ( 1 ) + + n - k ( k + 2 ) / 2 q k ( k + 2 ) ( 1 ) .
f ( x ) f * ( x ) + r ( x ) ,
r ( x ) = b 3 p 3 ( x ) f * ( x ) .
R ( x ) = 0 a r ( ξ ) d ξ .
r ( x ) = ( μ 3 - μ 3 * ) | 1 α 1 * α 2 * α 3 * α 1 * α 2 * α 3 * α 4 * α 2 * α 3 * α 4 * α 5 * 1 x x 2 x 3 | | 1 α 1 * α 2 * α 3 * α 1 * α 2 * α 3 * α 4 * α 2 * α 3 * α 4 * α 5 * α 3 * α 4 * α 5 * α 6 * | f * ( x ) .
N k = 1 α k * 0 x ξ k d Λ ( ξ ) = N ( ln x - λ * σ * - k σ * ) ,
R ( x ) = μ 3 - μ 3 * ( α 1 * ) 3 × - ρ 6 N 0 + ρ 2 ( ρ 4 + ρ 2 + 1 ) N 1 - ( ρ 4 + ρ 2 + 1 ) N 2 + N 3 ρ 12 ( ρ 2 - 1 ) 3 ( ρ 2 + 1 ) ( ρ 4 + ρ 2 + 1 ) ,
ρ 2 = 1 + [ q 2 ( 1 ) / n ]
μ 3 - μ 3 * ( α 1 * ) 3 = q 6 ( 1 ) n 2 ( 1 - 1 n ) .
R ( x ) [ n 7 / 6 q 12 ( 1 ) ] ,
N k = N 0 - k σ * N 0 ( 1 ) + 1 2 k 2 σ * 2 N 0 ( 2 ) - 1 6 k 3 σ * 3 N 0 ( 3 ) + ,
R ( x ) = - [ q 3 ( 1 ) / 6 n 1 2 ] N 0 ( 3 )
R ( x ) [ q 3 ( 1 ) / 6 n 1 2 ] ,