Cfrgtkky

I need to compute (or best approximate?) the following integral

$$int_0^infty int_0^infty (1 + alpha u)^{-1}(1 + v)^{-1} Phileft(frac{beta}{sqrt{gamma + uv}}right) text{d}u text{d}v,qquad gamma > 0, quad beta in mathbb{R}, quad alpha in mathbb{N}^*,$$
where $Phi(cdot)$ is the standard normal cumulative distribution function and $alpha$ is typically a large natural number. What strategy would you advise for this?

I don't know if this is helpful but I can also reformulate my problem (up to multiplicative constants) as computing the following expectation:

$$text{E}left{ Phileft(frac{beta}{sqrt{gamma + X^2Y^2}}right)right},qquad gamma > 0, quad beta in mathbb{R},$$
where
$$X sim text{Half-Cauchy}(0,1), qquad Y sim text{Half-Cauchy}(0,alpha^{-1/2}), quad alpha in mathbb{N}^* , (text{large}).$$

EDIT: I made a mistake in the first integral, it should be:

$$int_0^infty int_0^infty (1 + alpha x^2)^{-1}(1 + y^2)^{-1} Phileft(frac{beta}{sqrt{gamma + x^2y^2}}right) text{d}x text{d}y,quad gamma > 0, beta in mathbb{R}, alpha in mathbb{N}^*, text{(large)},$$
which now corresponds to the above expectation.

EDIT 2: Another way to solve my problem would be to characterise the difference (hopefully small?)
$$text{E}left{ Phileft(frac{beta}{sqrt{gamma + X^2Y^2}}right)right} - Phileft(frac{beta}{sqrt{gamma}}right),qquad gamma > 0, quad beta in mathbb{R},$$
would it be sensible to approximate the difference
$$Phileft(frac{beta}{sqrt{gamma + X^2Y^2}}right) - Phileft(frac{beta}{sqrt{gamma}}right),$$
with a Taylor series in zero or is this unreasonable given that the half-Cauchy distributions are heavy-tailed?

Under the conditions that $alpha, gamma, u,$ and $v$ are all positive,

$$beta/sqrt{gamma + uv} ge min(beta/sqrt{gamma}, 0) = delta gt -infty.$$

Therefore, because $Phi$ is a CDF for a distribution supported on $(-infty,infty),$ $$Phileft(frac{beta}{sqrt{gamma + uv}}right)ge Phi(delta) = epsilon gt 0.$$

Consequently

$$eqalign{
&int_0^infty int_0^infty (1+alpha u)^{-1}(1+v)^{-1} Phileft(frac{beta}{sqrt{gamma + uv}}right) mathrm{d}umathrm{d}v \
&ge
epsilonint_0^infty int_0^infty (1+alpha u)^{-1}(1+v)^{-1} mathrm{d}umathrm{d}v \
&= lim_{Mtoinfty}lim_{Ntoinfty}epsilonint_0^M (1+alpha u)^{-1}mathrm{d}uint_0^N (1+v)^{-1} mathrm{d}v \
&=frac{epsilon}{alpha} lim_{Mtoinfty}lim_{Ntoinfty} log(1 + Malpha)log(1 + N),
}$$

which diverges to $+infty.$

If approximation is OK, you could simulate the expected value through an average. A law of large numbers implies that the average converges to the expected value - provided the latter exists, which is however not obvious to me. I did simulate a few runs, though, and got very similar results each time, which is not indicative of an issue with heavy tails.

library(LaplacesDemon)



draws <- 1e6

alpha <- 3

gamma <- 1

beta <- 1

x <- rhalfcauchy(draws, scale=1)

y <- rhalfcauchy(draws, scale=1/sqrt(alpha))



mean(pnorm(beta/sqrt(gamma+x^2*y^2)))