3 Continuous limit

So far we have considered a finite set $\Omega$ and a fixed number of objects N. In reality, one often deals with a non-constant number of objects, so that N is itself a random variable. [For example, if the objects we are studying are galaxies and if $\Omega$ is a small field, the expected number of galaxies in our field will follow a Poissonian distribution with mean value $\rho A$, where $\rho$ is the density of detectable galaxies in the sky.] Clearly, when we observe a field $\Omega$ we will obtain a particular value for the number of objects N inside the field. However, in order to obtain more general results, it is convenient to consider an ensemble average, and take the number of observed objects as a random variable; the results obtained, thus, will be averaged over all possible values of N.

A way to include the effect of a variable N in our framework is to note that, although we are observing a small area on the sky, each object could in principle be located at any position of the whole sky. Hence, instead of taking N as a random variable and A fixed, we consider larger and larger areas of the sky and take the limit $A \rightarrow \infty$. In doing this, we keep the object density $\rho = N / A$ constant. It is easily verified that the two methods (namely A fixed and N Poissonian random variable with mean $\rho A$, or rather $A \rightarrow \infty$ with $\rho = A / N$ fixed), lead to the same results. In the following, however, we will take the latter scheme, and let A go to infinity; correspondingly we take $\Omega$as the whole plane.

From Eq. (16) we see that W(s) is proportional to 1/A. For this reason it is convenient to define the function

$$Q(s) = \int_\Omega \left[ \rm e^{-s w(\vec\theta)} - 1 \right] \, \rm d^2 \theta .$$ (18)

If A is finite, we have Q(s) = A W(s) - A. Thus we can write, in the limit $A \rightarrow \infty$,

$$Y(s) = \lim_{N \rightarrow \infty} \left[ 1 + \frac{Q(s) \rho}{N} \right]^{N-1} = \rm e^{\rho Q(s)} .$$ (19)

This equation replaces Eq. (17) when $N \rightarrow \infty$.

In order to further simplify the expression for the correcting factor C(w), we rewrite its definition as

$$C(w) = \rho \int_0^\infty \frac{\zeta_w(x)}{x} \, \rm dx ,$$ (20)

where x = y + w, and the function $\zeta_w$ is defined as

$$\zeta_w(x) = {\rm H}(x - w) p_y(x - w) .$$ (21)

Here ${\rm H}(x - w)$ is the Heaviside function at the position w, i.e.

$${\rm H}(x) = \left\{ \begin{array}{cc} 0 & \textrm{if}\ x < 0 , \\ 1 & \textrm{otherwise.} \end{array}\right.$$ (22)

The Laplace transform of $\zeta_w$ can be written as

$$Z_w(s) = \mathcal L[\zeta_w](s) = \rm e^{-ws} Y(s) .$$ (23)

In reality, for C(w) we need to evaluate an integral over $\zeta_w(x) / x$. From the properties of the Laplace transform we have

$$\mathcal L\bigl[ \zeta_w(x) / x \bigr](s) = \int_s^\infty Z_w(s') \, \rm ds' ,$$ (24)

and thus we find

 

$$C(w) =\rho \mathcal L\bigl[ \zeta_w(x) / x \bigr](0) = \rho \int_... ...s' =\rho \int_0^\infty \rm e^{-w s'} Y(s') \, \rm ds' = \mathcal L[\rho Y](w) .$$ (25)

This important result, together with Eqs. (18) and (19), can be used to readily evaluate the correcting factor. We note that, for our purposes, there is no need to evaluate the probability distribution p_y any longer. This prevents us from calculating any inverse Laplace transform.