# On The Decomposition Of Certain Random Variables

B. Ramachandran

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B. Ramachandran. On The Decomposition Of Certain Random Variables. Annales de l'ISUP, 1961, X (4), pp.267-273. hal-04092460

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# ON THE DECOMPOSITION OF CERTAIN RANDOM VARIABLES

B. RAMACHANDRAN

The Catholic University of America, Washington D. C.

## I - INTRODUCTION -

The subject of "Decomposition of Random Variables" has developed rapidly during the last twenty-five years. In 1936, H. Cramer [1] proved that if the sum of two independent random variables is Normally distributed, then so should each summand be. An analogous theorem for the Poisson variable was established by D. A. Raikov [7], and for the Binomial variable by N. A. Sapogov [8] and H. Teicher [9]. Recently, Yu. V. Linnik, in four fundamental papers, [2] to [5], has obtained extensive general results on the decomposition of certain "infinitely divisible laws". In particular, his paper [2] is devoted to the proof of the proposition that the composition of a Normal and a Poisson random variable can be factorised into such compositions only.

In this paper, we establish an analogous theorem on the composition of a Poisson and a Binomial random variable having the same maximum span. We also give examples to show that analogous results do not necessarily hold in such a case if the maximum spans are unequal, or in the case of the composition of a Normal and a Binomial-type random variable. We finally examine compositions of Binomial-type random variables.

## II - THE COMPOSITION OF A POISSON AND A BINOMIAL RANDOM VARIABLE HAVING THE SAME MAXIMUM SPAN -

We introduce certain definitions and a few simple lemmas. Lemmas 2.1 and 2.2 are known results.

###--------------------

(1) Research supported in part by the National Science Foundation under grant NSF-G-9968.
**Definition 2.1 :**

A distribution function is said to be a *lattice distribution* if it is purely discrete and its discontinuity-points form a proper or improper subset of a set of points of the form { a + kξ } , where a is a real number, ξ > 0, and k runs through integer-values.

The quantity ξ is called a *span* of the distribution. The largest of the numbers ξ for which the statement in the above definition holds is called the *maximum span* of the distribution (clearly, this exists provided there are at least two discontinuity points, that is, provided the distribution is not "improper"). Examples of lattice distributions are the classical Binomial and Poisson distributions, both of which have unit maximum span.

We assume that the reader is familiar with the concepts of the characteristic function (c.f.) of a random variable, and of the probability-generating function (p.g.f.) of a random variable taking only non-negative integer-values (see, for instance, [6], p. 20).

**Lemma 2.1.**

A necessary and sufficient condition that $f(t)$ should be the c.f. of a lattice distribution with a span $\xi$ is that $f(2\pi/\xi) = 1$.

For a proof of this simple result, the reader is referred to
[6], p. 27.

Since the c.f. of the sum of two independent random variables is the product of the c.f.'s of the summands, it follows at once from the above lemma that

**Lemma 2.2.**

If the sum of two independent random variables has a lattice distribution, then each of the summands also has a lattice distribution having the maximum span of their sum as a span.

**Lemma 2.3.**

$f(t) = (q+pe^{it})^n \exp[a(e^{it}-1)]$ can be a c.f. if and only if a $\ge$ 0
($n$ is a positive integer, $0 < p < 1$, $q = 1-p$) ; equivalently,
$(q + pq)^n \exp[a(z - 1)^n]$

can be a p.g.f. if and only if a $\ge$ 0.

Proof.

If $a = 0$, $f(t)$ is the product of the c.f.'s of a Binomial and of a Poisson random variable (the latter degenerates if $a = 0$), and hence is also a c.f.
Conversely, if $f(t)$ is a c.f., then we first note that $\underline{a}$ has to be real, since $f(-t) = f(t)$ for all real $t$. Hence we need only show that $\underline{a}$ cannot be negative. Now it is clear that

$$f(t) = \sum_{r=0}^{\infty} c_r e^{itr}$$

where

$$c_r = \begin{cases} e^{-a} \left[ q^n \frac{a^r}{r!} + \binom{n}{1} q^{n-1} p \frac{a^{r-1}}{(r-1)!} + \dots + \binom{n}{r} q^{n-r} p^r \right] & \text{for } r \le n, \\ e^{-a} \left[ q^n \frac{a^r}{r!} + \binom{n}{1} q^{n-1} p \frac{a^{r-1}}{(r-1)!} + \dots + p^n \frac{a^{r-n}}{(r-n)!} \right] & \text{for } r \ge n. \end{cases}$$

It is also clear that, for large $\underline{r}$, $c_r$ has the same sign as that of the dominant term in the expression for $c_r$, namely, the term $e^{-a} a^{r-n} p^{n/(r-n)}!$. Hence, if $a < 0$, then $c_r < 0$ for infinitely many values of $\underline{r}$, and so $f(t)$ cannot be a c.f. Hence the lemma.

We proceed to the main result.

### THEOREM 2.1.

The composition of a Poisson and a Binomial random variable having the same maximum span is factorisable only into such other compositions, also having the same maximum span.

### Proof.

Without loss of generality, we may take the parameter of the Poisson component as well as the common maximum span of the two components to be unity. Then, with the usual notation, the p.g.f. of the random variable Z under consideration is defined for all complex $z$, and is given by

$$C(z) = (q+pz)^n \cdot \exp(z-1) \quad (2.1)$$

If $Z = X + Y$, where X and Y are independent random variables, then, by Lemma 2.2, X and Y also have lattice distributions with unity as a span for each. Further, if $\underline{h}$ is a discontinuity point for Y, then, since the origin is the left-most discontinuity point for $Z = X + Y$, there cannot be a discontinuity point for X to the left of $-\underline{h}$: this is clear from the obvious relation:

$$\mathrm{Prob}(X) < -\underline{h}). \quad \mathrm{Prob}(Y = h) = \mathrm{Prob}(X < -h, Y = h) \le \mathrm{Prob}(X+Y < 0) = 0$$
on noting that Prob(Y=h) > 0 by assumption. This means that the distribution of X is "bounded to the left". A similar result holds for Y also. It is then clear that there is no loss of generality in taking the lattice-points of X and Y to be the non-negative integer-points and the origin to be a discontinuity-point for both random variables. We may then speak of the p.g.f.'s of X and Y. Let these be respectively :

$$A(z) = \sum_{r=0}^{\infty} a_r z^r, \quad \text{and} \quad B(z) = \sum_{r=0}^{\infty} b_r z^r \qquad (2.2)$$

where

$$a_r \ge 0, \quad b_r \ge 0, \quad a_o > 0, \quad \text{and} \quad b_o > 0 \qquad (2.3)$$

Obviously,

$$c_r = \text{Prob}(Z = r) = a_0 b_r + a_1 b_{r-1} + \dots + a_r b_0 \qquad (2.4)$$

so that, from (2.3), we have at once, for r = 0, 1, 2, ... ad inf.,

$$a_r \le b_0^{-1} c_r, \quad b_r \le a_0^{-1} c_r \qquad (2.5)$$

Hence,

$$\begin{align*} |A(z)| &= \left| \sum_{r=0}^{\infty} a_r z^r \right| \le \sum_{r=0}^{\infty} a_r |z|^r \le b_0^{-1} \left( \sum_{r=0}^{\infty} c_r |z|^r \right) \\ &= b_0^{-1} C(|z|)^n, \quad \text{from (2.1),} \\ &= b_0^{-1} (q+p|z|)^n \cdot \exp(|z|-1). \end{align*}$$

Hence, A(z), and similarly B(z), are entire functions of the complex variable $\bar{z}$, of order at most unity.

We then have, from (2.1), (2.2) and (2.4), that, for all $\bar{z}$,

$$A(z), B(z) = C(z) = (q+pz)^n \cdot \exp(z-1) \qquad (2.6)$$

It then follows from Hadamard's factorisation theorem (see, for instance, [10], p.250) that, since A(z) and B(z) are both entire functions of order at most unity,

$$A(z) = (q+pz)^l \cdot \exp(az+c)$$

$$B(z) = (q+pz)^m \cdot \exp(bz+d)$$
where a, b, c, d are (complex) constants, a+b = 1, l is a non-negative integer ≤ n, and m = n - l. Since A(1) = B(1) = 1, we have

$$A(z) = (q + pq)^l \cdot \exp[a(z - 1)]$$

$$B(z) = (q + pq)^m \cdot \exp[b(z - 1)]$$

Invoking Lemma 2.3, we immediately have a ≥ 0, b ≥ 0, thus proving the theorem.

### III - THE COMPOSITION OF A POISSON-TYPE AND A BINOMIAL-TYPE RANDOM VARIABLE WITH UNEQUAL MAXIMUM SPANS -

That a result analogous to Theorem 2.1 does not generally hold in the case of unequal maximum spans is illustrated by the following example. With the same notation as in Section 2, consider

$$C(z) = (1/2) (1 + z^3) \cdot \exp(z - 1)$$

Then

$$C(z) = A(z) \cdot B(z)$$

where

$$A(z) = (1 - z + z^2) \cdot \exp(z - 1), \quad \text{and} \quad B(z) = (1/2) (1 + z).$$

Of these, C(z) is the p. g. f. of the composition of a standard Poisson variable (i. e., having unit maximum span) and a Binomial-type variable with its maximum span equal to three units. A(z) has a power-series expansion with non-negative co-efficients, and A(1) being equal to unity, A(z) therefore represents a p. g. f. But, clearly, it is not the p. g. f. of the composition of a Binomial-type and a Poisson-type random variable. B(z) is, of course, the p. g. f. of a standard Binomial variable.

### IV - THE COMPOSITION OF A NORMAL AND A BINOMIAL-TYPE RANDOM VARIABLE -

That a result analogous to Theorem 2.1 does not generally hold in this case also is illustrated by the following example (this is perhaps the simplest example possible and was suggested by Dr. R. G. Laha) :

Consider the c. f.

$$f(t) = \cos t \cdot \exp(-t^2/4)$$
which is clearly the c.f. of the composition of a Binomial-type and
a Normal random variable. f(t) can be written as an infinite product

$$
f(t) = \prod_{k=0}^{\infty} P_k(t)
$$

where

$$
P_k(t) = \left[ 1 - \frac{4t^2}{(2k+1)^2 \pi^2} \right] \cdot \exp \left[ -2t^2 / (2k+1)^2 \pi^2 \right]
$$

Each P_k(t) is the c.f. of an indecomposable random variable (cf. [6], p.126). Since each P_k(t) is a continuous function of t and since $\prod_{k=1}^{\infty} P_k(t)$ is uniformly convergent in every finite interval of values of t, it is clear that this product is a continuous function of t. Also, $\prod_{k=1}^{\infty} P_k(t)$ is a c.f., being the product of a finite number of c.f.'s.

Hence, $\prod_{k=1}^{\infty} P_k(t)$, being the limiting function of a sequence of c.f.'s as also continuous (in fact, its continuity in some neighborhood of t = 0 is sufficient for our purposes), is itself a c.f., by the Lévy-Cramér continuity theorem. P_c(t) is a c.f., as we have already seen, and it is obviously not the c.f. of the composition of a Normal and a Binomial-type random variable, being indeed indecomposable. This completes the argument in justification of the example.

V - COMPOSITIONS OF BINOMIAL-TYPE VARIABLES -

It is easy to verify that the composition of any finite number
of Binomial-type random variables having the same maximum span
(which may be taken as unity, without loss of generality) can be
decomposed only into such other compositions. That this need not
be the case if the maximum spans are unequal is exemplified by
the following (using the notation of Section 2) : take

$$
C(z) = \frac{1 + 8z^3}{9} \cdot \left(\frac{1+z}{2}\right)^2
$$

$$
A(z) = \frac{1 + 2z}{3}
$$

$$
B(z) = \frac{(1+z)^2 \cdot (1 - 2z + 4z^2)}{12}
$$

B(z) corresponds to a non-Binomial-type random variable.
VI - CONCLUSION -

The problem of under what conditions a result analogous to
Theorem 2.1 will hold in the cases discussed in Sections 3 to 5
still remains open.

REFERENCES

1. H. CRAMÉR - "Über eine Eigenschaft der Normalen Verteilungs-funktion", *Math. Zeits.*, Vol. 41, pp. 405-414, 1936.

2. Yu. V. LINNIK - "On factorising the composition of a Gaussian and a Poisson law", *Teoriia Ver. i ee Prim.*, Vol. 1, No. 1, pp. 34-59, 1957.

3. Yu. V. LINNIK - "General theorems on the factorisation of infinitely divisible laws, I", *Teoriia Ver. i ee Prim.*, Vol. 3, No. 1, pp. 3-40, 1958.

4. Yu. V. LINNIK - "General theorems on the factorisation of infinitely divisible laws, II", *Teoriia Ver. i ee Prim.*, Vol. 4, No. 1, pp. 55-85, 1959.

5. Yu. V. LINNIK - "General theorems on the factorisation of infinitely divisible laws, III", *Teoriia Ver. i ee Prim.*, Vol. 4, No. 2, pp. 150-171, 1959.

6. E. LUKACS - *Characteristic Functions*, Griffin's Statistical Monographs and Courses No. 5, Charles Griffin and Co. Ltd., London, 1960.

7. D. A. RAIKOV - "On the decomposition of the Poisson law", *Dokl. Akad. Nauk. SSSR*, Vol. 14, pp. 9-12, 1937.

8. N. A. SAPOGOV - "The stability problem for a theorem of Cramer", *Izvest. Akad. Nauk. SSSR (Ser. Matem.)*, Vol. 15, pp. 205-218, 1951.

9. H. TEICHER - "On the factorisation of distributions", *Ann. Math. Stat.*, Vol. 25, pp. 769-774, 1954.

10. E. C. TITCHMARSH - *Theory of Functions*, Oxford University Press, Oxford, England, 1939.