7.1 Distribution approximations  (Page 3/3)

Before examining a general approximation procedure which has significant consequences for later treatments, we consider some illustrative examples.

An m-procedure for discrete approximation

If X is bounded, absolutely continuous with density functon f _X , the m-procedure tappr sets up the distribution for an approximating simple random variable. An interval containing the range of X is divided into a specified number of equal subdivisions. The probability mass for each subinterval is assigned to the midpoint. If $dx$ is the length of the subintervals, then the integral of the density function over the subintervalis approximated by $f_X\left(t_i\right)dx$ . where t _i is the midpoint. In effect, the graph of the density over the subinterval is approximated by a rectangle of length $dx$ and height $f_X\left(t_i\right)$ . Once the approximating simple distribution is established, calculations are carried out as for simple random variables.

Because of the regularity of the density and the number of approximation points, the result agrees quite well with the theoretical value.

The next example is a more complex one. In particular, the distribution is not bounded. However, it is easy to determine a bound beyond which the probability is negligible.

The general approximation procedure

We show now that any bounded real random variable may be approximated as closely as desired by a simple random variable (i.e., one having a finite set of possible values). For the unboundedcase, the approximation is close except in a portion of the range having arbitrarily small total probability.

We limit our discussion to the bounded case, in which the range of X is limited to a bounded interval $I=[a,b]$ . Suppose I is partitioned into n subintervals by points t _i , $1\le i\le n-1$ , with $a=t_0$ and $b=t_n$ . Let $M_i=[t_{i-1},t_i)$ be the i th subinterval, $1\le i\le n-1$ and $M_n=[t_{n-1},t_n]$ (see [link] ). Now random variable X may map into any point in the interval, and hence into any point in each subinterval M _i . Let $E_i=X^{-1}\left(M_i\right)$ be the set of points mapped into M _i by X . Then the E _i form a partition of the basic space Ω . For the given subdivision, we form a simple random variable X _s as follows. In each subinterval, pick a point $s_i,t_{i-1}\le s_i<t_i$ . Consider the simple random variable ${\displaystyle X_s=\sum _{i=1}^n{s}_i{I}_{E_i}}$ .

This random variable is in canonical form. If $\omega \in E_i$ , then $X\left(\omega \right)\in M_i$ and $X_s\left(\omega \right)=s_i$ . Now the absolute value of the difference satisfies

Since this is true for each ω and the corresponding subinterval, we have the important fact

By making the subintervals small enough by increasing the number of subdivision points, we can make the difference as small as we please.

While the choice of the s _i is arbitrary in each M _i , the selection of $s_i=t_{i-1}$ (the left-hand endpoint) leads to the property $X_s\left(\omega \right)\le X\left(\omega \right)\forall \omega$ . In this case, if we add subdivision points to decrease the size of some or all of the M _i , the new simple approximation Y _s satisfies

To see this, consider $t_i^{*}\in M_i$ (see [link] ). M _i is partitioned into $M_i^{\text{'}}\bigvee M_i^{\text{'}\text{'}}$ and E _i is partitioned into $E_i^{\text{'}}\bigvee E_i^{\text{'}\text{'}}$ . X maps $E_i^{\text{'}}$ into $M_i^{\text{'}}$ and $E_i^{\text{'}\text{'}}$ into $M_i^{\text{'}\text{'}}$ . Y _s maps $E_i^{\text{'}}$ into t _i and maps $E_i^{\text{'}\text{'}}$ into $t_i^{*}>t_i$ . X _s maps both $E_i^{\text{'}}$ and $E_i^{\text{'}\text{'}}$ into t _i . Thus, the asserted inequality must hold for each ω By taking a sequence of partitions in which each succeeding partition refines the previous (i.e. adds subdivision points) in such a way that themaximum length of subinterval goes to zero, we may form a nondecreasing sequence of simple random variables X _n which increase to X for each ω .

The latter result may be extended to random variables unbounded above. Simply let N th set of subdivision points extend from a to N , making the last subinterval $[N,\infty )$ . Subintervals from a to N are made increasingly shorter. The result is a nondecreasing sequence $\{X_N:1\le N\}$ of simple random variables, with $X_N\left(\omega \right)\to X\left(\omega \right)$ as $N\to \infty$ , for each $\omega \in \Omega$ .

For probability calculations, we simply select an interval I large enough that the probability outside I is negligible and use a simple approximation over I .