Let S = { k 1 , k 2 , … } S = \{ {k_1},{k_2}, \ldots \} be an increasing sequence of positive integers. We call S S strongly ergodic if for every measure preserving transformation T T on a probability space ( Ω , F , P ) (\Omega ,\mathcal {F},P) and every f ∈ L 1 ( Ω ) f \in {L_1}(\Omega ) we have lim n → ∞ ( 1 / n ) ∑ j = 1 n f ( T k j ω ) = P f ( ω ) {\lim _{n \to \infty }}(1/n)\sum \nolimits _{j = 1}^n {f({T^{kj}}\omega ) = Pf(\omega )} a.e. where P f Pf is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence S S to be strongly ergodic and provide examples.

Let G be a l.c.a. group and {T g } be a representation of G such that each T g is a measure-preserving transformation on some probability space (Ω, J^, P).Let {μ n } be a sequence of probability measures on (?.We are interested in the a.e.convergence or summability of I f(T g w)dμ n (9), for feL^Ω).Some examples and counterexamples are given, and some partial results are obtained.

Let {Xn} n =1/∞ be a sequence of random variables with partial sumsSn, and let {ie241-1} be the σ-algebra generated byX1,…,Xn. Letf be a function fromR toR and suppose {ie241-2}. Under conditions off and moment conditions on theX'ns, we show thatSn/n converges a.e. (almost everywhere). We give several applications of this result.

Let (Ω,ℱ,P) be a probability space and let {itXn(ω)}n=1 be a sequence of i.i.d. random vectors whose state space isZm for some positive integerm, where Z denotes the integers. Forn = 1, 2,... letSn(ω) be the random walk defined by\(S_n (\omega ) = \sum\limits_{j = 1}^n {X_j (\omega )}\). ForxεZm andα∈Um, them-dimensional torus, let\(\left\langle {\alpha ,x} \right\rangle = e^{2\pi i} \sum\limits_{j = 1}^m {\alpha _j x_j }\). Finally let\(\phi (\alpha ) = E\{ \left\langle {\alpha ,X_1 (\omega )} \right\rangle \}\) be the characteristic function of the X's.

Let (Ω,f,P) be a probability space and letT be a measure-preserving weak mixing transformation. We define a large class of sequences of integers calledp-sequences, such that iff∈L 1 there exists a set Ω′⊂Ω of probability one and for ω∈Ω′ we have \(\mathop {\lim }\limits_{n \to \infty } \frac{1}{n}\sum\limits_{f = 1}^n {f\left( {T^k ,\omega } \right)} = \int_\Omega {f\left( \omega \right)} \mu \left( {d\omega } \right)\) for everyp-sequence {kn}.

Let $a = \{f(\cdot, \theta): \theta \in J\}, J$ an interval, be a family of univariate probability densities (with respect to Lebesgue measure) on an interval I. First, a necessary and sufficient condition is proved for $\mathcal{a}$ to be identifiable whenever $\mathcal{a} \subset C_0(J)$, the class of continuous functions on $J$ vanishing at $\infty$. If $f_G$ is a $G$-mixture of the densities in $a$ with $G$ unknown, an estimator $G_n$ based on $f_G$ and $\mathscr{B} = \{f(x, \bullet): x \in I\}$ is provided such that $G_n \rightarrow_w G$ under certain conditions on $a$. If $X_1, \cdots, X_n$ are i.i.d. random variables from $f_G$, an estimator $\hat{G}_n$ is provided such that $G_n(X_1, \cdots, X_n, \cdot) \rightarrow_w G(\bullet)$ almost surely under certain conditions on $a$ and $G$. Furthermore, it is shown that $|f_{G_n}(x) - f_G(x)| \rightarrow 0$ a.s. and in $L_2$ with rates like $O(n^{-C}) (C > 0)$ under certain conditions on the density estimator $\hat{f}_G(x)$ involved in the definition of $\hat{G}_n$. The conditions of various theorems are verified in the case of location parameter and scale parameter families of densities.

Let G G be an LCA group and h h a Hilbert space, and let T ( g ) T(g) be a function on G G into the contractions on H H . Let { σ n } \{ {\sigma _n}\} be a sequence of probability measures on G G . Under suitable conditions on T ( g ) T(g) and the sequence { σ n } \{ {\sigma _n}\} we prove the strong convergence of the sequence T n = ∫ T ( g ) σ n ( d g ) {T_n} = \smallint T(g){\sigma _n}(dg) . In certain cases we identify the limiting operator.

Let Z Z be the group of integers and Z ¯ \bar Z its Bohr compactification. A sequence of probability measures { μ n , n = 1 , 2 , … } \{ {\mu _n},n = 1,2, \ldots \} defined on Z Z is said to be ergodic provided μ n {\mu _n} converges weakly to μ ¯ \bar \mu , the Haar measure on Z ¯ \bar Z . Let A n ⊂ Z , n = 1 , 2 , … {A_n} \subset Z,n = 1,2, \ldots and define μ n {\mu _n} by μ n ( B ) = | A n ∩ B | / | A n | {\mu _n}(B) = |{A_n} \cap B|/|{A_n}| where | B | |B| is the cardinality of B B . Then it is easy to show that if | A n ∩ A n + k | / | A n | → 1 |{A_n} \cap {A_n} + k|/|{A_n}| \to 1 for every k ∈ Z k \in Z , then μ n {\mu _n} is ergodic. Let 0 ≤ p k ≤ 1 0 \leq {p_k} \leq 1 . In this paper we construct (random) sequences { μ n } \{ {\mu _n}\} which are ergodic, and such that lim ( | A n ∩ A n + k | / | A n | ) = p k \lim (|{A_n} \cap {A_n} + k|/|{A_n}|) = {p_k} , for every k ∈ Z k \in Z .

Let G G and H H be locally compact σ \sigma -compact abelian groups, A \mathcal {A} a mapping from G G to H H , and { μ n } n = 1 ∞ \{ {\mu _n}\} _{n = 1}^\infty a sequence of measures on G G . We define the notions: “ A \mathcal {A} is a uniform distribution with respect ot { μ n } \{ {\mu _n}\} ” and “ A \mathcal {A} is a strong uniform distribution". We give a number of examples of these notions and derive some general individual ergodic theorems for measure-preserving transformations with discrete spectrum.

For a givenn-tuple of non-negative numbers (p 0,p 1,...,p n−1) whose sum is equal to unity let μ(t) denote the probability that Σ j = 1/∞ X j /n j ≦t, where the independent random variablesX j assume the values 0,1,...,n−1 with probabilitiesp 0,p 1,...,p n−1 respectively. For mostn-tuples we obtain upper and lower bounds on |û(m)|; these estimates involve then-ary representation ofm, or in some cases of 2m, so that a very simple and explicit characterization of the sequences on whichû(m) approaches zero can be given. In particular, for the Cantor middle-third measure, corresponding to the triple (1/2, 0, 1/2), the following criterion is obtained.û(m) approaches zero on a sequenceT of integers if and only if Ω(2m) approaches infinity onT, where Ω(k) is the sum of the following three quantities associated with the ternary representation ofk: the number of runs of zeros, the number of runs of twos and the number of ones. The results obtained are easily extended to the case when then-tuple varies withj (subject to certain mild restrictions).

Let (Θ, ℱ, μ) be a probability space andT a 1-1, onto, measure-preserving transformation. Necessary and sufficient conditions are given forT to be mixing, in terms of union of iterates of sets.

Let {X k ,k=1,2,…} be a sequence of independent binomial variables, with\(P\{ X_k = 1\} = 1 - P\{ X_k = 0\} = P_k \cdot Let Y = \sum\limits_{k = 1}^\infty {X_k /2^k and \hat \mu (t)} be\) the Fourier transform of the distribution ofY. Finally denote lim [P k − 1/2] byδ. We haveTheorem. \((4/\pi )\delta \leqq \overline {\mathop {\lim }\limits_{x \to \infty } } \left| {\hat \mu (t)} \right| \leqq 2\delta \)

0. Introduction. To the best of the authors' knowledge almost all of the work that has been done until the present on confidence intervals and confidence sets of fixed precision has either concerned independent (usually identically distributed) observations, or has been asymptotic in character. In this paper we treat the problem of fixed length confidence intervals for the parameters of a discrete m-dependent stationary Gaussian process. Our main result is somewhat depressing; namely, that if m is unknown (i.e., the possible distributions consist of all m-dependent such processes for all m) such estimation is impossible. In fact it is impossible in a rather small subclass of these processes. In this area there are, however, quite a few surprises. For example, the authors had conjectured that the main difficulty would arise in attempting to distinguish a case of independent observations with large mean and small variance from the case of 0 mean highly correlated observations with large variance. In both cases one would see a large first observation followed by a number of observations close by, and it appeared difficult to arrive at a stopping rule in which one could distinguish these two cases. Our intuition appeared to be justified when we were able to show (Theorem 1) that for one class in which independence-large mean-small variance and high dependence-large variance cases were both included, there is no J-stage scheme for fixed length confidence interval estimation of the mean whose last J -1 sample sizes are determined by differences of values observed in previous stages. Recall that in Stein's two sample scheme the second sample size is determined by the first stage sample variance