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Paper [3] shows by example that the lognormal distribution is not determined by its moments. It is referenced in Feller’s second volume (Feller 1971) and has become a widely cited result.
The best-known result on rates of convergence in the central limit theorem is undoubtedly that of A.C. Berry and C.-G. Esseen, which describes the rate in the case of finite third moments. In particular, if X,X1,X2, … are independent and identically distributed random variables for which 1 $$E\, |\, X\, |^3 <\, \infty,$$ $${\rm if}\ E(X)=0\ {\rm and}\ E(X^2)=1,\...
In one of his first few papers, [4], Chris gives necessary and sufficient conditions for the convergence of the series 1 $$\begin{array}{*{20}c} {\sum\limits_{n = 1}^\infty {n^k P\left( {S_n \leq x} \right),} } & { - \infty < x < \infty,} \\ \end{array}$$ for integers k = 0, 1, 2, …, and for a similar series with nk replaced by e...
When Chris Heyde returned to Australia in September 1968 to take up aposition as Reader in Ted Hannan’s teaching Department of Statistics in the-thenSchool of General Studies at the Australian National University, there was still astrong trend of research into population genetics emanating from Pat Moran’spurely research department in the Institute of Advanced Studies, a directionsynthesized in Moran...
Summary In this paper it is established that the lognormal distribution is not determined by its moments. Some brief comments are made on the set of distributions having the same moments as a lognormal distribution.
Let Xi, i=1, 2, 3,… be a sequence of independent and identically distributed random variables and write Sn = X1+X2+… +Xn. If the mean of Xt is finite and positive, we have Pr(Sn ≦ x) → 0 as n → ∞ for...
Let $$\{X_j, j = 1, 2, 3, \ldots\}$$ be a sequence of independent, non-degenerate random variables and write $$S_n = \sum^{n}_{j=1} X_j$$ Under quite a diverse variety of conditions we may obtain $${\rm Pr}(S_n < n^p x)\rightarrow \frac{1}{2}$$ as $$n \rightarrow \infty$$ for all $$x, - \infty < x < \infty$$ , and some real $$\mathcal{P} \geqslant 0$$ . For example,...
Let the random variables $$X_i, i = 1, 2, 3, \ldots$$ be independent and identically distributed. Write $$S_n = \sum^{n}_{i=1} X_i \, {\rm and}\, Z_n = S_n B^{-1}_n - A_n$$ for normed and centered sums. The classical central limit formulation is for $${\rm Pr} (Z_n \leqq x_n)$$ where xn = 0(1) as $$n \rightarrow \infty$$ and thus gives only trivial...
Let $$X_i, i = 1, 2, 3, \ldots,$$ be a sequence of independent and identically distributed random variables with law $$\mathcal{L}(X)$$ and write $$S_n = \sum\nolimits^{n}_{i=1} X_i$$ . Let $$x_n, n = 1, 2, 3, \ldots,$$ be a monotone sequence of positive numbers with $$x_n \rightarrow \infty \,{\rm as}\, n \rightarrow \infty$$ such that $$x_n^{-1} S_n \rightarrow_P 0$$...
Let $$X_i, i = 1, 2, 3, \ldots$$ be a sequence of independent and identically distribution random variables with $$X_i=\sigma^2<\infty$$ and $$EX_i=0.$$ Write $$S_n=\sum^n_{i=1}Xi,\quad n\geqq1.$$ Then, $$F_n(x) = {\rm Pr} (S_n \leqq \sigma \sqrt{n}x) \rightarrow \Phi(x) = \frac{1}{\sqrt{2\pi}}\int\limits^{x}_{\infty}e^{-\frac{1}{2}u^2} du$$ as n → ∞, and the rate...
Summary In this paper, the precise asymptotic behaviour of the large deviation probability is found in the case where the random variables are attracted to a non-normal stable law. This extends previous work of the same author in which only the order of magnitude of the large deviation probability was found.
Let Xi, i = 1,2,3, … be a sequence of independent and identically distributed random variables with $$\mathcal{L}(X)$$ and write $$S_n = \sum {_{i = 1}^n X_i. }$$ If $$EX=0$$ and $$EX^2=\sigma^2<\infty$$ , the law of the iterated logarithm (Hartman and Wintner [1]) tells us that $$\Pr ({\mathop {\lim \sup }\limits_{n \to \infty }({2\sigma...
Let $$\{X_i, i=1, 2, 3, \ldots\}$$ be a sequence of independent and identically distributed random variables and write $$S_n=\sum\nolimits^n_{i=1} X_i,\: n\geqq1.$$ It is well known that $$\Pr ({\mathop {\lim \sup }\limits_{n \to \infty } \!({2 n\:\log \:\log \:n})^{ - {1}/ {2}} \!{S_n } = 1}) = 1$$ If and only if $$EX_i=0,\: EX^2_i=1$$ (Hartman and Wintner [6] obtained...
Let {Xii = 1, 2, 3, …} be a sequence of independent and identically distributed random variables for which EXi = 0, var Xi = σ2, E|Xi|2 +a = b < ∞, a > 0. C[0, 1] is the space of continuous real-valued functions on the interval...
Let Xii =1,2,3,... be a sequence of independent and identically distributed random variable which belong to the domain of attraction of a stable law of index α. Wrie $$S_0=0,\quad S_n\sum\nolimits^{n}_{i}=1 \,{\mathbf{X}}_i, n\geqq 1,$$ and $$M_n={\rm max}_{0\leqq k \leqq n} S_k.$$ In the case where the Xi are such that $$\sum\nolimits^{\infty}_{1}\,...
Summary This paper is concerned with the use of the ℒ1 and ℒ∞ metrics in a study of certain properties and implications of convergence rates in the central limit theorem for sums of independent and identically distributed random variables which belong to the domain of attraction of the normal distribution. Also, some general convergence rate results on the ℒ∞ metric obtained under the assumption of...
Let Z0 = 1, Z1, Z2... denote a super-critical Galton–Watson process whose non-degenerate offspring distribution has probability generating function $$F\left(s\right)=\sum\nolimits^{\infty}_{{j}=0}s^j \mathbf{P}{\rm r}\left(\mathbf{Z}_1=j\right), 0\leqq s \leqq 1,$$ where 1 <m =EZ1 >∞. The Galton–Watson process evolves in such that the generating...
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