Conditions/real definition of a kernel function (regression)

Every kernel function definition I have found has that it integrates to 1 and that the expectation is 0. However there is a third condition which I've seen a couple of times (but not everywhere) which is that the variance is finite. What does this mean for the kernel? (My guess is that it can't tend to infinity)

Also, what would it mean if the integral of K^2 is finite?

Thanks

Re: Conditions/real definition of a kernel function (regression)

The **most widely used kernels** are symmetric and have a finite second moment, which means that the variance is finite (less than infinity).

Hopefully that clears something up?

Edit: You can also think of finite variance as a regularity condition that needs to be satisfied (probably always for the purpose of your class).

Re: Conditions/real definition of a kernel function (regression)

Thanks abender,

Does this mean there is no set definition for a kernel, just some conditions that "most" kernels satisfy?

Re: Conditions/real definition of a kernel function (regression)

Your text or professor may have a specific definition (i.e., requirements) for a kernel function.

Finite variance is an example of a regularity condition. Regularity conditions are annoyingly vague, and depend on context. For example, a regularity condition for a, say, function, is that the function is differentiable (or smooth) enough. In one context, that may mean that the function can be differentiated 4 times.

Bottom line, ask your instructor. He or she may have a definition of choice for kernel.