Ratio limit theorem for renewal processes

We consider a renewal process which models a cumulative shock model that fails when the accumulation of shocks up-crosses a certain threshold. The ratio limit properties of the probabilities of non-failure after n cumulative shocks are studied. We establish that the ratio of survival probabilities converges to the probability that the renewal epoch equals zero. This limit holds for any renewal process, subject only to mild regularity conditions on the individual shock random variable. Precisions on the rates of convergence are provided depending on the support structure and the regularity of the distribution. Arguments are provided to highlight the coherence between this new results and the pre-existing results on the behavior of summands of i.i.d. real random variables.

💡 Research Summary

This paper presents a rigorous mathematical analysis of the asymptotic behavior of survival probabilities in a cumulative shock model based on a renewal process. The model considers a system subjected to a sequence of i.i.d. non-negative random shocks, {X_i}, representing damage increments. The system fails at the first time the cumulative sum S_n = Σ_{i=1}^n X_i exceeds a fixed critical threshold x > 0. The central object of study is the sequence of survival probabilities c_{n,x} = P(S_n ≤ x) = P(failure time > n), and specifically the asymptotic properties of the ratio of consecutive terms, r_n = c_{n+1,x} / c_{n,x}.

The authors establish a general “Ratio Limit Theorem,” proving that under mild regularity conditions, this ratio converges to the probability that an individual shock equals zero, i.e., lim_{n→∞} r_n = P(X_1 = 0). This result holds for a wide variety of renewal processes. The primary innovation and depth of the work lie in the precise characterization of the rates of convergence of this ratio, which are shown to depend critically on the local behavior of the distribution function F of X_1 near zero.

To systematically handle this dependence, the paper introduces a classification of distributions into five classes (C1-C5) based on their support and regularity at the origin:

• C1: Distributions with support bounded away from zero (leading to trivial, zero probabilities for large n).
• C2: Discrete distributions where zero is an isolated atom and there exists a smallest positive jump size x_min > 0.
• C3: Discrete distributions where zero is an accumulation point of the support, and F(t) - F(0) = O(t^α) for some α > 0 as t→0+.
• C4: Absolutely continuous distributions with a density f that is regularly varying with index α-1 at zero and has a monotone derivative nearby.
• C5: Mixed distributions with an absolutely continuous part satisfying C4 conditions and a discrete part with atoms bounded away from zero.

For discrete distributions (C2, C3), the analysis uses probabilistic decompositions and combinatorial arguments. For class C2, Proposition 2.2 not only confirms convergence to P(X_1=0) but provides an exact asymptotic equivalence: r_n - P(X_1=0) ~ P(X_1=0) * M_x / n, where M_x is an integer representing the maximum number of positive jumps possible without crossing x. Proposition 2.3 further shows how individual atom probabilities P(X_1=y) can be asymptotically recovered from the c_{n,x} sequence under certain gap conditions. For class C3, Theorem 2.1 establishes the same limit but with a less explicit convergence rate due to the possibility of infinitely many small jumps.

The analysis for absolutely continuous distributions (C4) shifts to the properties of the n-fold self-convolution density f^{*n} of S_n. A key technical lemma (Proposition 2.4) states that for a C4 density, after a sufficiently large number of convolutions (n ≥ N_x), the convolved density f^{*n} becomes non-decreasing on any fixed interval