${\bi T},{\bi T}^{\bf -1}$ is not standard

An elementary proof of de Finetti’s theorem

Introduction. These results are generalizations of a theorem of Chung [I ] on the limiting distribution of the number of crossings of a value, c, by the successive partial sums of a sequence of independent random variables, and of theorems by Chung and Erdos [3 ] on the lower limits of such sums. Two of these theorems concern the case of independent, equidistributed random variables whose distributions have an absolutely continuous component (Case A). The others are for binomial variates (Case B). We extend the results of Case A to sums of independent random variables whose distributions need not be the same, and those of Case B to sums of independent and equidistributed random variables of the lattice type. Let {X } be a sequence of independent random variables whose c.d.f.'s, { Fn(x) }, need not be the same. We use the usual notations:

Until now, we mostly dealt with a single random variable at a time and used it to model a random quantity, such as the waiting time for a customer to enter a store or the number of deaths due to horse kicks. Occasionally the problem gave rise to a sequence of independent random variables like in the coupon collector’s problem. We studied tools that help us with the analysis of single random variables and sequences of independent random variables. However, in applications we frequently encounter sequences of random variables that are not independent. Andrey Markov was one of the firsts who gave a systematic analysis of such sequences in the early 1900s. Today we call these sequences Markov chains.

In the present work the results of K. L. Chung [2] concerning the maximum partial sums of sequences of independent random variables are obtained for a weaker condition. The method employed in the proof is analogous to the one used by Chung with the difference that, instead of Esseen’s approximations involving third moments, we use Berry’s approximations involving only second moments.

SummaryGnedenko's classical work [1] on the limit of the distribution of the maximum of a sequence of independent random variables is extended to the distribution of the maximum of a sequence of random variables defined on a finite Markov chain.

Let ${X_n;n\geq 0}$ be a sequence of random variables. We consider its geometrically weighted series $\xi(\beta)=\sum_{n=0}^\infty \betaX_n$ for $0<\beta < 1$. This paper proves that $\xi (\beta)$ can be approximated by $\sum_{n=0}^\infty \beta^n Y_n$ under some suitable conditions, where ${Y_n; n \geq 0}$ is a sequence of independent normal random variables. Applications to the law of the iterated logarithm for $\xi(\beta)$ are also discussed.

Convergence for weighted sums of Lüroth type random variables

It is well known that the sample mean, based on a sequence of independent random variables with common distribution, is a weakly consistent estimator as well as a strongly consistent estimator for the population mean. The first property is the famous weak law of large numbers (Khinchin), the second property is the strong law of large numbers (Kolmogorov). Both laws are key results of modern probability theory. Similar properties hold for more complicated sequences of random variables, for example, for sequences satisfying the stationarity assumption or for martingale sequences. These extensions are of crucial importance for statistical applications in ergodic theory and survival analysis.

We study the class of random entire functions given by power series, in which the coefficients are formed as the product of an arbitrary sequence of complex numbers and two sequences of random variables. One of them is the Rademacher sequence, and the other is an arbitrary complex-valued sequence from the class of sequences of random variables, determined by a certain restriction on the growth of absolute moments of a fixed degree from the maximum of the module of each finite subset of random variables. In the paper we prove sharp Wiman–Valiron’s type inequality for such random entire functions, which for given p∈(0;1) holds with a probability p outside some set of finite logarithmic measure. We also considered another class of random entire functions given by power series with coefficients, which, as above, are pairwise products of the elements of an arbitrary sequence of complex numbers and a sequence of complex-valued random variables described above. In this case, similar new statements about not improvable inequalities are also obtained.

A version of the law of the iterated logarithm is proved for sequences of independent random variables which satisfy the central limit theorem in such a way that the convergence of the appropriate moment-generating functions to that of the standard normal distribution occurs at a particular rate. Kolmogorov's law of the iterated logarithm is a corollary of this theorem which, unlike Kolmogorov's result, does not require boundedness of the random variables. Some iterated logarithm results for weighted averages of independent random variables are shown to follow from the main result. Moreover, some applications to sequences of independent, generalized Gaussian random variables are provided.

Burkholder [3] gave a simple necessary and sufficient condition, in terms of concentration functions, for independent sequences of random variables to have the Stein property. Here we find sufficient conditions for the Stein property without assuming that the random variable sequence is independent. Our conditions are also in terms of concentration functions, but in our case they are conditional concentration functions which specialize to those used by Burkholder. For some of our results, the sequence of random variables may be quite arbitrary; however, we usually assume it to be a martingale difference sequence satisfying certain regularity conditions.

In this paper we study a reflected AR(1) process, i.e. a process (Zn)n obeying the recursion Zn+1= max{aZn+Xn,0}, with (Xn)n a sequence of independent and identically distributed (i.i.d.) random variables. We find explicit results for the distribution of Zn (in terms of transforms) in case Xn can be written as Yn−Bn, with (Bn)n being a sequence of independent random variables which are all Exp(λ) distributed, and (Yn)n i.i.d.; when |a|&lt;1 we can also perform the corresponding stationary analysis. Extensions are possible to the case that (Bn)n are of phase-type. Under a heavy-traffic scaling, it is shown that the process converges to a reflected Ornstein–Uhlenbeck process; the corresponding steady-state distribution converges to the distribution of a normal random variable conditioned on being positive.

Let $\{\xi_1,\xi_2,\ldots\}$ be a sequence of independent random variables, and $\eta$ be a counting random variable independent of this sequence. We consider conditions for $\{\xi_1,\xi_2,\ldots\}$ and $\eta$ under which the distribution function of the random sum $S_{\eta}=\xi_1+\xi_2+\cdots+\xi_{\eta}$ belongs to the class of consistently varying distributions. In our consideration, the random variables $\{\xi_1,\xi_2,\ldots\}$ are not necessarily identically distributed.

Suppose that $p_{n} = 10 {\rm log}_{10} [10^{x_{1}/10} + \ldots + 10^{x_{n}/10}]$,$ where {X n } is a sequence of independent random variables. The main result of this paper shows that under very general conditions on the sequence {X n }, the power sums P n will be asymptotically normally distributed. This result supports a commonly used normal approximation, and shows why many physical quantities obtained by power addition of random variables tend to be normally distributed in dB.

Let $X_1, X_2,\cdots$ be a sequence of independent random variables, each with zero mean and finite variance. Define $S_n = \sum^n_{i=1} X_i, s_n^2 = E(S_n^2), t_n^2 = 2 \log \log s_n^2$ and $\Lambda = \lim \sup_{n\rightarrow\infty} S_n/(s_n t_n)$. Suppose that $|X_n| \leqq c_n s_n$ a.s. for all $n$ and some real sequence $\{c_n\}$ and that $s_n \rightarrow \infty$, and let $\nu = \lim \sup_{n\rightarrow\infty} t_nc_n$. By Kolmogorov's law of the iterated logarithm, $\Lambda = 1$ if $\nu = 0$. Egorov proved that $0 \leqq \Lambda < \infty$, in every case. In this paper it will be shown that, if $\nu < \infty$, then $0 < \Lambda \leqq 1 + \sum^\infty_{j=3} \nu^{j-2}/j!$. A similar result for certain classes of unbounded random variables will also be presented.