Completely Monotone Multisequences, Symmetric Probabilities and a Normal Limit Theorem

Proceedings Mathematical Sciences

J C Gupta¹ 

[1] Indian Statistical Institute, , SJS Sansanwal Marg, New Delhi 0 0, India Corresponding Address: , Mirdha Tola, Budaun 0, India$$

关键词: Moment spaces;    completely monotone multisequences;    normal limit.;   

DOI  :  

学科分类：数学（综合）

来源: Indian Academy of Sciences

【 摘 要 】

Let G𝑛, 𝑘 be the set of all partial completely monotone multisequences of order 𝑛 and degree 𝑘, i.e., multisequences 𝑐𝑛(𝛽1, 𝛽2,$ldots$ ,𝛽k), 𝛽1, 𝛽2,$ldots$ ,𝛽𝑘 = 0, 1, 2,$ldots$ ,𝛽1 + 𝛽2 + $cdots$ + 𝛽𝑘 ≤ n, 𝑐𝑛(0,0,$ldots$ ,0) = 1 and $(-1)^{𝛽_0}𝛥^{𝛽_0}$ 𝑐𝑛(𝛽1, 𝛽2,$ldots$ ,𝛽𝑘)≥ 0 whenever 𝛽0 ≤ 𝑛-(𝛽1 + 𝛽2 +$cdots$ +𝛽𝑘) where 𝛥 𝑐𝑛(𝛽1, 𝛽2,$ldots$ ,𝛽)=𝑐𝑛(𝛽1+1, 𝛽2,$ldots$ ,𝛽𝑘)+ 𝑐𝑛(𝛽1,𝛽2+1,$ldots$ ,𝛽𝑘)+$cdots$ + 𝑐𝑛(𝛽1, 𝛽2,$ldots$ ,𝛽𝑘+1)-𝑐𝑛(𝛽1,𝛽2,$ldots$ ,𝛽𝑘)$. Further, let $prod_{n,k}$ be the set of all symmetric probabilities on ${0, 1, 2,ldots ,k}^{n}$. We establish a one-to-one correspondence between the sets G𝑛, 𝑘 and $prod_{n, k}$ and use it to formulate and answer interesting questions about both. Assigning to G𝑛, 𝑘 the uniform probability measure, we show that, as 𝑛 → ∞, any fixed section {𝑐𝑛(𝛽1, 𝛽2,$ldots$ ,𝛽𝑘), 1 ≤ $sum 𝛽𝑖≤ 𝑚}, properly centered and normalized, is asymptotically multivariate normal. That is, $left{sqrt{left(inom{n+k}{k}ight)}(𝑐𝑛(𝛽1, 𝛽2,ldots ,𝛽𝑘)-c_0(𝛽1, 𝛽2,ldots ,𝛽𝑘), 1≤ 𝛽_1+𝛽2+cdots +𝛽_k≤ might}$ converges weakly to MVN[0,𝛴𝑚]; the centering constants 𝑐0(𝛽1, 𝛽2,$ldots$ ,𝛽𝑘) and the asymptotic covariances depend on the moments of the Dirichlet $(1, 1,ldots ,1; 1)$ distribution on the standard simplex in 𝑅𝑘.

【 授权许可】

Unknown