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MATHEMATICS
O. S. BERLYAND, I. M. NAZAROV, A. Ya. PRESSMAN
\(i^n\operatorname{erfc}\)- OR THE COMPLEX GAUSS–POISSON DISTRIBUTION
(Presented by Academician N. N. Bogolyubov, 25 VI 1962)
By the \(i^n\operatorname{erfc}\) (en effect) or complex Gauss–Poisson distribution we shall mean the probability distribution for the occurrence of the number of events \(n\), obeying a Poisson law whose parameter is itself a random variable following the truncated normal distribution \(N(x,a,\sigma)\) with abscissa of the truncation point equal to zero:
\[ P(n)= \frac{2}{1+\operatorname{erf}\dfrac{a}{\sigma\sqrt{2}}}\, \frac{1}{\sqrt{2\pi}\sigma n!} \int_0^\infty x^n e^{-x-(x-a)^2/2\sigma^2}\,dx = \]
\[ = \frac{e^{y^2/4-a}}{1+\operatorname{erf}\dfrac{a}{y}}\, y^n i^n\operatorname{erfc}\left(\frac{y}{2}-\frac{a}{y}\right), \tag{1} \]
where \(y=\sigma\sqrt{2}\).
The \(i^n\operatorname{erfc}\)-distribution describes a sufficiently broad class of random events, for example: a) fluctuations in the number of elementary particles emitted according to a Poisson law from radiation sources whose intensity or concentration varies randomly in time or space; b) the distribution of the number of crystallization centers over the volume of a melt with a random distribution of the concentration of the substance (for example, in magmatic foci); c) the distribution of output data for discrete devices that register events obeying a Poisson law, whose counting efficiency is a random variable (in particular, for counters of various kinds of radiation operating not entirely stably), etc.
We compute the mathematical expectation and variance of the \(i^n\operatorname{erfc}\)-distribution using the basic identity
\[ \frac{e^{y^2/4-a}}{1+\operatorname{erf}\dfrac{a}{y}} \sum_{n=0}^{\infty} y^n i^n\operatorname{erfc}\left(\frac{y}{2}-\frac{a}{y}\right) \equiv 1. \tag{2} \]
Rewriting (2) in the form
\[ \sum_{n=0}^{\infty} y^n i_n\left(\frac{y}{2}-\frac{a}{y}\right) = e^{a-y^2/4}\left(1+\operatorname{erf}\frac{a}{y}\right) \tag{3} \]
(where, for brevity, the notation \(i_n(x)\) has been used instead of \(i^n\operatorname{erfc}(x)\)) and differentiating (3) with respect to \(y\), we find the first and second moments
\[ M_1=\sum_{n=1}^{\infty} nP(n) = a+\frac{y}{\sqrt{\pi}}\, \frac{e^{-a^2/y^2}}{1+\operatorname{erf}\dfrac{a}{y}}; \tag{4} \]
\[ M_2=\sum_{n=1}^{\infty} n^2P(n) = D^2+M_1^2 = a+a^2+\frac{y^2}{2} + (a+1)\frac{y}{\sqrt{\pi}}\, \frac{e^{-a^2/y^2}}{1+\operatorname{erf}\dfrac{a}{y}}, \tag{5} \]
where \(D^2\) is the variance of the distribution.
The calculation of the probabilities \(P(n)\) can be carried out from tables of the functions \(i^n \operatorname{erfc} x\) (1) or by the approximate formulas (2).
For \(a>10\) the \(i^n\operatorname{erfc}\)-distribution can, with satisfactory accuracy, be approximated by the truncated normal distribution
\[ \frac{2}{1+\operatorname{erf}\dfrac{n^*}{D\sqrt{2}}}\, \frac{1}{D\sqrt{2\pi}}\, e^{-(n-n^*)^2/2D^2}, \tag{6} \]
where \(n^*\) is the abscissa of the mode.
The position of the mode of the distribution can be determined as follows. Let \(n^*=m\); then
\[ y^{m-1} i_{m-1}\left(\frac{y}{2}-\frac{a}{y}\right) < y^m i_m\left(\frac{y}{2}-\frac{a}{y}\right); \tag{7} \]
\[ y^{m+1} i_{m+1}\left(\frac{y}{2}-\frac{a}{y}\right) < y^m i_m\left(\frac{y}{2}-\frac{a}{y}\right), \tag{8} \]
where \(\dfrac{y}{2}-\dfrac{a}{y}\geq 0\). Using the consequence of (8)
\(i_{m+1}(x)/i_m(x)<1/y\) and the inequality (see (?))
\[ \frac{i_{m-1}(x)}{i_m(x)} \leq 2x+\frac{m+1}{x}, \tag{9} \]
we find a lower estimate for the mode \(m=n^*\) for \(x=y/2-a/y>0\):
\[ n^*>x(y-2x)-2. \]
On the other hand, from the relations
\[ \frac{d}{dx}\bigl(\ln i_m(x)\bigr) = -2x-2(m+1)\frac{i_{m+1}(x)}{i_m(x)}, \qquad i'_m(x)=-i_{m-1}(x), \]
formula (9) and the consequence of (7) \(i_m(x)/i_{m-1}(x)>1/y\), we obtain an upper estimate for \(m=n^*\) when \(x>0\), i.e. \(y>\sqrt{2a}\),
\[ n^*< \frac{(2x^2+1)2a}{(4x-y)y}. \]
Thus,
\[ x(y-2x)-2<n^*< \frac{(2x^2+1)2a}{(4x-y)y}. \tag{10} \]
For \(y\gg 2\sqrt{a}\) we have \(n^*\simeq a\); \((a\gg 2)\).
In the case of small \(x\), using (7) and (8), expanding the function \(i^n\operatorname{erfc}(x)\) in a Taylor series and retaining two terms, we obtain
\[ \frac{i_n(0)-i_{n-1}(0)x}{i_{n-1}(0)-i_{n-2}(0)x} > \frac{1}{y}, \qquad \frac{i_{n+1}(0)-i_n(0)x}{i_n(0)-i_{n-1}(0)x} < \frac{1}{y}; \]
whence, a fortiori,
\[ \frac{i_n(0)}{i_{n-1}(0)-i_{n-2}(0)x} > \frac{1}{y}, \qquad \frac{i_{n+1}(0)-i_n(0)x}{i_n(0)} < \frac{1}{y}. \]
Next, since
\[ i_n(0)=\frac{1}{2^n\Gamma(1+n/2)}, \]
we shall have
\[ \frac{2\Gamma(1+(n+1)/2)}{\Gamma(1+n/2)} > \frac{y}{1+xy}, \qquad \frac{2\Gamma(1+n/2)}{\Gamma(1+(n-1)/2)} < y+nx. \tag{11} \]
Let us estimate the value of the mode of the distribution in the case of a negative argument, i.e. when \(0>-(a/y-y/2)=-x\). Using the recurrence relation
\[ 2m i_m(-x)=2x i_{m-1}(-x)+i_{m-2}(-x), \]
write the ratio in the form of the continued fraction
\[ \frac{i_m(-x)}{i_{m-1}(-x)} = \frac{x}{m} + \frac{1}{2m}\cdot \frac{1}{ \dfrac{x}{m-1}+\dfrac{1}{2(m-1)}\cdot \dfrac{1}{ \dfrac{x}{m-2}+\cdots \dfrac{1}{ \dfrac{x}{2}+\dfrac{1}{2\cdot2}\cdot \dfrac{1}{\dfrac{x}{1}+\dfrac{1}{2}\gamma(x)} } }. \tag{12} \]
where \(\gamma(x)=i_1(-x)/i_0(-x)=2e^{-x^2}/\sqrt{\pi}(1+\operatorname{erf}x)\).
From (12) it follows that, for all \(x\),
\[ \frac{1}{2k}\, \frac{1}{ 2x+\dfrac{(2k-2)!!}{(2k-1)!!} } \leq \frac{i_{2k}(-x)}{i_{2k-1}(-x)} \leq \frac{(2k-1)!!}{(2k)!!}\,\gamma(x)+x; \tag{13} \]
\[ \frac{1}{2k+1}\, \frac{1}{ 2x+\dfrac{(2k-1)!!\cdot 2}{(2k)!!}\cdot\dfrac{1}{\gamma(x)} } \leq \frac{i_{2k+1}(-x)}{i_{2k}(-x)} \leq \frac{(2k)!!}{(2k+1)!!}\,\frac{\gamma(x)}{2}+x. \tag{14} \]
For large \(k\), using Stirling’s formula, we obtain
\[ \frac{(2k+1)!!}{(2k)!!} = \frac{(2k+1)(2k)!}{2^{2k}(k!)^2} \sim \frac{2\sqrt{k}}{\sqrt{\pi}}; \qquad \frac{(2k)!!}{(2k-1)!!} = \frac{2^{2k}(k!)^2}{(2k)!} \sim \sqrt{\pi k}. \tag{15} \]
Without loss of generality, one may assume that \(n^*=2k\); then from (7), (8), (13), and (14) we shall have
\[ \frac{\gamma(x)}{2}\,[y-2x(2k+1)] \leq \frac{(2k+1)!!}{(2k)!!}; \qquad \frac{(2k)!!}{(2k-1)!!} < \frac{1}{(1/y-x)\gamma(x)}, \tag{16} \]
and for large \(k\)
\[ \frac{\gamma(x)\sqrt{\pi}}{4}\,[y-2x(2k+1)] < \sqrt{k} < \frac{1}{(1/y-x)\gamma(x)\sqrt{\pi}}. \tag{17} \]
Formulas (16) and (17) are suitable for sufficiently small \(x\); in particular, for \(x=0\) \((a=y^2/2)\), from (16) we have
\[ \frac{y}{\sqrt{\pi}} < \frac{(2k+1)!!}{(2k)!!}; \qquad \frac{(2k)!!}{(2k-1)!!} < \frac{y\sqrt{\pi}}{2}, \tag{18} \]
and for large \(k\) it follows from (18) that \(\sqrt{k}=y/2\), whence
\[ n^*=2k=\frac{y^2}{2}=a. \tag{19} \]
If, however, \(x=a/y-y/2\) is sufficiently large, then from (12) it follows that
\[ \frac{x}{m} \leq \frac{i_m(-x)}{i_{m-1}(-x)} \leq \frac{x}{m}+\frac{m-1}{m}\frac{1}{2x}, \]
whence, putting \(n^*=m\) and using (7) and (8), we shall have
\[ xy-1\leq n^*\leq \frac{x-1/2x}{1/y-1/2x}, \tag{20} \]
i.e., for sufficiently large \(x\),
\[ n^*\simeq xy=a-\frac{y^2}{2}. \]
Institute of Applied Geophysics
Academy of Sciences of the USSR
Received
20 VI 1962
REFERENCES
- O. S. Berlyand, R. I. Gavrilova, A. P. Prudnikov, Tables of integral functions, error functions and Hermite polynomials, Minsk, 1961.
- O. S. Berlyand, A. Ya. Pressman, DAN, 140, No. 1 (1961).