A Statistical Problem Arising in the Theory of Detection of Signals in the Presence of Noise in a Multi-Channel System and Leading to Stable Distribution Laws

The following statistical problem arising in the theory of radar is solved. There are two hypotheses concerning joint densities of random variables ₁, ₙ \ gathered A: PA (x₁, , xₙ) = ₈ = ₁ⁿ {2xᵢ de^ - xᵢ^{{2 / d} } }, \\ B: PB (x₁, , xₙ) = 1n₉ = ₁ⁿ {2xⱼ {d + d}e^ - xd^{{2 {d + d}} } } ₈ ₉ {2xᵢ de^ - xᵢ^{{2 / d} } }, \\ gathered \ where d > 0, d > 0. Let = { d / d} > 0. Denote by ₙ (F, D) such a number that there are no tests for distinguishing between hypotheses A and B with error probabilities which are smaller than F and D for ₙ (F, D) and that tests exist for > ₙ (F, D). It is proved that for n = const, D = const and F 0\ ₙ (F, D) = n + {1 / F} { {1 / D}} - 1 + o (1). asymptotic formula containing stable distributions for ₙ (F, D), in which F = const, D = const and n, is derived.