Abstract Generated abstract
This paper proves a transfer theorem for sums of a random number of independent, identically distributed random variables in a triangular array. Under convergence of a fixed-index sum, convergence of the normalized counting variable, and proportional growth of the normalizing constants, the limiting characteristic function of the random sum is represented as an integral mixture of powers of the fixed-sum limit characteristic function. The result is then connected with infinite divisibility via the Lévy representation and used to identify broad classes of characteristic functions, including extensions of Linnik-type forms, gamma-mixture cases, and examples arising from geometric and uniform distributions of the random index.
Full Text
UDC 519.21
MATHEMATICS
Academician of the Academy of Sciences of the Ukrainian SSR B. V. GNEDENKO, HUSSEIN FAHIM
ON A CERTAIN TRANSFER THEOREM
A large number of mathematical and applied problems lead to the necessity of summing independent random variables in a random number and of studying limit distributions for such sums. After the papers \((^{1,2})\) this topic attracted the attention of many researchers. Here we shall note only the works \((^{3-5})\). In \((^{4,5})\) there is a fairly complete bibliography.
Consider a sequence of integer-valued random variables \(\{\nu_n\}\) and an array of random variables \(\xi_{nk}\) with two entries. We shall assume that for each \(n\) the variables \(\xi_{nk}\) are identically distributed and independent in the aggregate. Suppose further that, for each \(n\), the variable \(\nu_n\) is independent of the variables of the sequence \(\{\xi_{nk}\}\).
Theorem. If there exists a sequence \(\{k_n\}\) such that, as \(n \to \infty\):
a)
\[ \mathbf{P}\left\{\sum_{k=1}^{k_n} \xi_{nk}<x\right\}\to \Phi(x); \]
b) with a suitable choice of constants \(c_n\)
\[ \mathbf{P}\left\{\frac{\nu_n-k_n}{c_n}<x\right\}\to A(x), \]
where \(\Phi(x)\) and \(A(x)\) are distribution functions;
c) \(c_n/k_n\to r\ (0<r<\infty)\),
then the distributions of the sums
\[ S_{\nu_n}=\xi_{n1}+\xi_{n2}+\ldots+\xi_{n\nu_n} \]
as \(n\to\infty\) converge to a limit \(\Psi(x)\). The characteristic function \(\psi(t)\) of the limiting distribution is equal to
\[ \psi(t)=\int_{0}^{\infty}[\varphi(t)]^z\,dA^*(z), \]
where \(\varphi(t)\) is the characteristic function of the distribution \(\Phi(x)\), and \(A^*(z)=A(y-1/r)\).
Proof. The characteristic function of the sum \(S_{\nu_n}\) is equal to
\[ \varphi_n(t)=\sum_{j=0}^{\infty}p_{nj}(f_n(t))^j, \]
where
\[ p_{nj}=\mathbf{P}\{\nu_n=j\}, \qquad f_n(t)=\int_{-\infty}^{\infty} e^{itx}\,dF_n(x). \]
Put
\[ A_n(x)=\mathbf{P}\{\nu_n<x\}. \]
It is obvious that
\[ \varphi_n(t)=\int_0^\infty f_n^x(t)\,dA_n(x). \]
Let now
\[ \overline{A}_n(y)=\mathbf P\left\{\frac{\nu_n-k_n}{c_n}<y\right\}=A_n(c_ny+k_n). \]
In this notation
\[ \varphi_n(t)=\int_{-k_n/c_n}^{\infty} f^{yc_n+k_n}(t)\,d\overline{A}_n(y) = \int_{-k_n/c_n}^{\infty} \left[f^{k_n}(t)\right]^{yc_n/k_n+1}\,d\overline{A}_n(y). \]
According to the assumptions of the theorem and the known assumptions on passing to the limit under the integral sign, in every finite interval of \(t\)
\[ \psi(t)=\lim_{n\to\infty}\varphi_n(t)=\int_{-r^{-1}}^\infty \varphi^{ry+1}(t)\,dA(y). \]
If now we denote \(z=ry+1\), then we arrive at the assertion of the theorem.
Since, according to a well-known theorem of A. Ya. Khinchin ((6), p. 197), \(\varphi(t)\) is the characteristic function of an infinitely divisible distribution and, with a suitable choice of \(F(x)\) and \(\{k_n\}\), one can obtain any infinitely divisible characteristic function, then
\[ \varphi(t)=e^{v(t)}, \]
where \(\gamma\) is an arbitrary real constant, \(G(x)\) is a nondecreasing function of bounded variation, and
\[ v(t)=i\gamma t+\int_{-\infty}^{\infty}\left\{e^{itx}-1-\frac{itx}{1+x^2}\right\}\frac{1+x^2}{x^2}\,dG(x). \]
Since
\[ \psi(t)=\int_0^\infty e^{iy(-iv(t))}\,dA^*(y)=a(-iv(t)), \]
where
\[ a(t)=\int_0^\infty e^{itx}\,dA^*(x), \]
we conclude that if \(a(t)\) is the characteristic function of a random variable taking only nonnegative values, and \(v(t)\) is the logarithm of the characteristic function of an infinitely divisible distribution, then the function
\[ \psi(t)=a(-iv(t)) \]
is characteristic. Moreover, according to a remark of W. Feller ((7), p. 646), if the function \(A^*(y)\) is infinitely divisible, then \(\psi(t)\) is the characteristic function of an infinitely divisible distribution.
Example 1. If \(\nu_n\) has a geometric distribution, then
\[ A^*(z)=1-e^{-z}\qquad (z\geqslant 0). \]
The functions
\[ \psi(t)=\int_0^\infty e^{-z(1-v(t))}\,dz=\frac{1}{1-v(t)}, \]
according to what has been proved, are characteristic and infinitely divisible, since the distribution \(1-e^{-z}\) is infinitely divisible.
Example 2. Characteristic and at the same time infinitely divisible are all functions
\[ \psi(t)=\frac{1}{1+c|t|^\alpha\left\{1+i\beta \frac{t}{|t|}\omega(t,\alpha)\right\}+i\gamma t}, \]
where the constants \(c>0,\ -1\le \beta \le 1,\ 0<\alpha\le 2,\ -\infty<\gamma<\infty\), and
\[ \omega(t,\alpha)= \begin{cases} \tg \dfrac{\pi}{2}\alpha, & \text{for } \alpha\ne 1,\\[6pt] \dfrac{2}{\pi}\ln |t|, & \text{for } \alpha=1. \end{cases} \]
The proof that the functions
\[ \psi(t)=\frac{1}{1+|t|^\alpha}, \qquad 0<\alpha\le 2, \]
are characteristic was given in the work of Yu. V. Linnik \((^8)\). Our example supplements this result in two directions.
Example 3. All functions
\[ \psi(t)=\left(\frac{1}{1-v(t)}\right)^\alpha, \]
where \(\alpha>0\), and \(v(t)\) is the logarithm of the characteristic function of an infinitely divisible distribution, are characteristic and at the same time infinitely divisible.
For the proof it suffices to take
\[ A^*(z)=\int_0^z \frac{x^{\alpha-1}}{\Gamma(\alpha)}e^{-x}\,dx \qquad (z\ge 0). \]
Example 4. If \(\nu_n\) is uniformly distributed on the interval \((0;2k_n)\), then as \(n\to\infty\)
\[ \mathbf{P}\left\{\frac{\nu_n-k_n}{2k_n}<x\right\}\to \begin{cases} 0, & \text{for } x\le -0.5,\\ 0.5x, & \text{for } -0.5\le x\le 0.5,\\ 1, & \text{for } x>0.5. \end{cases} \]
Thus, all functions
\[ \psi(t)=\int_0^1 e^{zv(t)}\,dz=\frac{e^{v(t)}-1}{v(t)}, \]
where \(v(t)\) is the logarithm of the characteristic function of an infinitely divisible distribution, are characteristic (but not necessarily infinitely divisible).
In particular, the functions
\[ \psi(t)=\frac{1-e^{-|t|^\alpha}}{|t|^\alpha} \]
are characteristic for all constants \(\alpha,\ 0<\alpha\le 2\).
Moscow State University
named after M. V. Lomonosov
Received
22 III 1969
CITED LITERATURE
\(^1\) H. Robbins, Proc. Nat. Acad. Sci. U.S.A., 34, 162 (1948).
\(^2\) H. Robbins, Bull. Am. Math. Soc., 54, 1151 (1948).
\(^3\) R. L. Dobrushin, UMN, 10, issue 2 (64), 157 (1955).
\(^4\) W. Richter, Math. Nachr., 29, Heft 5–6, 347 (1965).
\(^5\) S. Guiașu, Studii și cercetări Matem., 19, fasc. 7, 971 (1967).
\(^6\) B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Moscow–Leningrad, 1949.
\(^7\) V. Feller, An Introduction to Probability Theory and Its Applications, 2, Moscow, 1967.
\(^8\) Yu. V. Linnik, Ukr. Mat. Zh., 5, issues 2–3, 207, 247 (1953).