Today KKK showed me the following formula:
There exists a constant such that
for all . Here the integral is either over the unit sphere or over the ball (of course the corresponding constants are different). In this note we explain this result in a probabilistic way and give more examples.
Exercise: Find the constants in both cases.
An obvious generalization of the above formula is as follows. Here we write for .
Proposition 1. Let be a Borel probability measure on such that
Then for all , we have
Proof. We calculate directly:
We observe that the converse is also true (why?).
To get KKK’s formula from Proposition 1, simply check that the condition is satisfied. This is an exercise in integration. (Hint: Use symmetry!)
Now we translate the above into a probabilistic framework.
Let be a real random variable. The expectation of is defined as
provided that the integral exists absolutely. The variance of is defined as
provided the expectation exists absolutely. See Wikipedia for more details about these definitions.
Definition. Let and be real random variables with finite variances. The covariance between and is defined as
We say that and are uncorrelated if .
Note that is bilinear and (it is basically the -inner product). We remark that if and are independent, then they are uncorrelated. The converse is not true. Statistically, implies that there are no “linear relationships” between and . This statement can be made precise using orthogonal projection. In this note, all random variables have expectation . In this case the formula becomes
Let be a random vector on , i.e. each is a random variable. The distribution of is the (Borel) probability measure on defined by
where is any Borel set in . We also note the following basic result: If is a real measurable function on , then
provided the integrals exist. (Actually the existence of one integral implies the existence of the other.)
With the above notations, we can rephrase Proposition 1 as follows:
Proposition 2 (Probabilistic version of Proposition 1). Consider a random vector on with distribution . Suppose that for all and
Then for all , we have
Proof. We simply note that
The idea is that checking is equivalent to checking that certain random variables are uncorrelated, and sometimes the probabilistic way is more intuitive.
All these are quite trivial. Now we give some examples.
Example 3. For all ,
To see this, note that is the the normal distribution on with mean vector and covariance matrix (identity). It follows that if has this distribution, then are independent and identically distributed with mean and variance .
Example 4. To get KKK’s identity for the spherical integral, we let be uniformly distributed on the unit sphere . Note that have the same distribution (but they are independent). By Proposition 2 (and scaling), we only need to show that and are uncorrelated for . Of course this can be interpreted as an integration problem; here we give a probabilistic trick.
Now is uniformly distributed on . We can construct a strictly positive random variable , independent of , such that
is standard normal as in Example 3. (The idea is similar to that of the Box–Muller transform in statistics. Roughly speaking, we first choose the length of , and then randomly pick a direction.)
Then, for , by independence of and (this follows from a standard theorem in statistics) we have
On the other hand, by construction of we get
Since , it follows that as well. By scaling appropriately we can use Proposition 3 (note here that ).
Example 5. For KKK’s identity for the volume integral (over the ball), we can use a similar argument as in Example 4, but we use another so that has the correct distribution.