bibkey: “kabluchko2009products” authors: “Zakhar Kabluchko” year: 2009 title: “Limiting Distributions for Sums of Independent Random Products” doi: “10.48550/arXiv.0904.4127” url: “https://arxiv.org/abs/0904.4127v2” claim: “For exponentially many independent products with a fixed increment law, the mean-normalized sum converges in probability to one half at the first critical parameter.” strata_touched: [] license: “citation-only” triage: “anchor”

Limiting Distributions for Sums of Independent Random Products

The inspected source is arXiv:0904.4127v2, revised 24 November 2009; v1 was submitted 27 April 2009. The PDF has 31 pages. Theorem 5, section 1.4, page 6, gives the critical limit one half for the normalized sum of independent random products. Its assumptions (1) and (4) require exp(-c n) N_n -> 1, a fixed nondegenerate increment law, and finite exponential moments at every nonnegative parameter. The first critical parameter is c_1 = phi'(1) - phi(1), where this paper’s phi is the log moment generating function. Lemma 4 in section 4.4 treats truncated exponential moments under an exponential tilt.

The critical loss of mean is a literature-attested random-energy phenomenon. This result does not directly cover the parity-kernel mixture: its likelihood components share observations, its amplitude may vary arbitrarily with dimension, and its path observations have temporal dependence. The parity result supplies a distinct-location joint limit under a double likelihood tilt and an exact cross-orientation overlap bound. Those are the model-specific repo-derived parts, rather than a claim that critical loss of mean itself is new.