An Integral Equation in Probability

Abstract

The integral equation

$$ \begin{gathered} \hfill \\ f(x) = \frac{1} {{\sqrt {2\pi } }}\int\limits_\infty ^\infty {e^{ - x^2 /2(\beta _0 + \beta _{1v^2 } } } )\frac{{f(v)}} {{\sqrt {\beta _0 + \beta _{1v^2 } } }}dv' \hfill \\ \end{gathered} $$

where the quantities β ₀ and β ₁ are positive constants, makes its appearance in the discussion of ARCH models in statistics. The original model was proposed by Engle [1], and Pantula [2] was the first to write down the equation in this connection. The function f is a distribution function and as such is to have the properties that f is nonnegative and that its integral is unity. The questions of interest (apart from existence and uniqueness of a solution) are how the number of moments depends on the parameter β _l and what is the asymptotic behavior of f (x) as |x| → ∞.