Testing Fiscal Sustainability via Strong Convergence of Gaussian Random Variables

Abstract

We prove, via the Borel-Cantelli lemma, that for every sequence of Gaussian random variables the combination of convergence in expectation and decreasing variances at fractional-polynomial rate implies strong convergence. This result has an important consequence for macroeconomic stochastic infinite-horizon models: The almost sure transversality condition (i.e., fiscal sustainability with probability one) is satisfied if (a) the discounted levels of net liabilities are Gaussian-distributed with fractional-polynomially decaying variances and (b) their means converge to zero. If (a) holds but (b) fails, the transversality condition will be almost surely violated. Hence, (a) and (b) constitute a test for almost sure fiscal sustainability.

Keywords:

• Almost sure convergence
• Borel-Cantelli lemma
• Fiscal sustainability
• Gaussian random variables
• Stochastic infinite-horizon models
• Transversality condition

Mathematics Subject Classification:

• 60F15
• 60G15
• 62P20
• 60F20
• 91B82

Acknowledgment

This work was supported by a research grant from the German Academic Exchange Service (DAAD).

Notes

The integrand is bounded by e on [0, 1], so it is enough to consider the integral . However, is decreasing in p, therefore we only need to show that this integral is finite for arbitrarily small p, e.g. for all p of the form p = 2^−N with N ∈ ℕ. Let, therefore, N ∈ ℕ, and apply the change of variable formula to z = x ^2−N , which gives

Now, every integral of the form is finite, as can be seen through an inductive argument based on integration by parts. Therefore, is finite for all a > 0 and N ∈ ℕ, whence we may conclude from our previous deliberations that is finite for all a, p > 0.