Concerning a Certain Probability Problem

Let ξ ₁ ,ξ ₂ , ⋯ be a sequence of independent $(0,1)$ normal random variables and let \[ {gathered} λ _1^2 = λ _2^2 = ⋯ λ n_1^2 , \\ λ n_1 + 1^2 + λ n_1 + 2^2 = ⋯ = λ n_1 + n_2^2 , \\ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ \\ {gathered} \] be a sequence of positive numbers such that \[ λ _1^2 > λ n_1 + 1^2 > ⋯ { and }∑_k λ _k^2 <infty . \] We prove the following asymptotic formula for the distribution of the random variable η = ∑ₖ λ ₖ² ξ ₖ²: \[ {gathered} { P}\{ η x\} = 1 - F_η (x) = {K} {{Γ ( {{{n_1 }}{2}} )}}( {{x}{{2λ _1^2 }}} )( n_1/ 2)-1 e- x/2λ _1^2 [ 1 + ε _1 (x) ], \\ p_η (x)={K}{{( {2λ _1^2 } )n_1/ 2 Γ ( {{{n_1 }}{2}} )}}x^{( {{{h_1 } {/ { {{h_1 } 2}} . -} 2}} ) - 1} e^{{{ - x} {/ { {{ - x} {2λ _1^2 ( {1 + ε _2 (x)} )}}} . -} {2λ _1^2}}} ( {1 + ε _2 (x)} ), \\ {gathered} \] where ε ⱼ (x) → 0 as x → ∞ and \[ K = ∏k = n_1 + 1^∞ {( {1 - {{λ _k^2 }}{{λ _1^2 }}} )- 1 .} \]