To each of the items $1,2,\cdots, N$ in a finite population there is associated a variate value. The population is sampled by successive drawings without replacement in the following way. At each draw the probability of drawing item $s$ is proportional to a number $p_s > 0$ if item $s$ remains in the population and is 0 otherwise. Let $\Delta(s; n)$ be the probability that item $s$ is obtained in the first $n$ draws and let $Z_n$ be the sum of the variate values obtained in the first $n$ draws. Asymptotic formulas, valid under general conditions when $n$ and $N$ both are "large", are derived for $\Delta(s; n), EZ_n$ and $\operatorname{Cov}(Z_{n_1}, Z_{n_2})$. Furthermore it is shown that, still under general conditions, the joint distribution of $Z_{n_1}, Z_{n_2},\cdots, Z_{n_d}$ is asymptotically normal. The general results are then applied to obtain asymptotic results for a "quasi"-Horvitz-Thompson estimator of the population total.

Optimality criteria formulated in terms of the power functions of the individual tests are given for problems where several hypotheses are tested simultaneously. Subject to the constraint that the expected number of false rejections is less than a given constant $\gamma$ when all null hypotheses are true, tests are found which maximize the minimum average power and the minimum power of the individual tests over certain alternatives. In the common situations in the analysis of variance this leads to application of multiple $t$-tests. In that case the resulting procedure is to use Fisher's "least significant difference," but without a preliminary $F$-test and with a smaller level of significance. Recommendations for choosing the value of $\gamma$ are given by relating $\gamma$ to the probability of no false rejections if all hypotheses are true. Based upon the optimality of the tests, a similar optimality property of joint confidence sets is also derived.

Two theorems on the asymptotic normality of linear combinations of functions of order statistics are given. Theorem 1 requires a "smooth" scoring function but the underlying df need not be continuous even and can also depend on the sample size. Theorem 2 allows general scoring functions but places additional restrictions on the df. Applications included.

Let $X_1, X_2, \ldots, X_N$ be independent identically distributed random variables with common continuous distribution function $F$. Designate by $\mathscr{J}$ a nonempty set of subsets of the integers $\{ 1, 2,\ldots, N\}$ and by $\mathscr{Y} = \mathscr{Y}(\mathscr{J})$ the mapping which assigns to each set $I \in \mathscr{J}, I = \{t_1, t_2, \ldots, t_k\}$ the partial sum $\sum{t_j \in I}X_{t_j}$. Define the random variable $N = N(\mathscr{J})$ as the number of positive sums in the range of $\mathscr{Y}. N(\mathscr{J})$ has been shown to be distribution free when $F$ is the distribution function of a symmetric random variable if $\mathscr{J} = \{1,2, \ldots, N\}$ or $\mathscr{J} = \text{power set of} \{1,2, \ldots, N\}$. Several other nontrivial examples of this phenomenon have been discovered--all by different methods. This paper presents a unified method that derives all previously known results, provides a constructive method for obtaining infinitely many essentially different sets $\mathscr{J}$ with this property, and finally provides a powerful necessary condition on any such set $\mathscr{J}$ that yields a complete characterization of those sets $\mathscr{J}$ for which $N(\mathscr{J})$ is distribution free and $\mathscr{J}$ contains all $k$ element subsets of $\{1,2, \ldots, N\}$ where $k = 2,3, \ldots, N - 1$.

This paper considers the possible limit laws for a sequence of normalized extreme order statistics (maximum, second maximum, etc.) from a stationary strong-mixing sequence of random variables. It extends the work of Loynes who treated only the maximum process. The maximum process leads to limit laws that are the same three types that occur when the underlying process is a sequence of independent random variables. The results presented here show that the possible limit laws for the $k$th maximum process $(k > 1)$ from a strong-mixing sequence form a larger class than can occur in the independent case.

Let $X$ be a random vector, taking values in $p$-dimensional Euclidean space $\mathscr{E}^p$ with density $f(x; \theta)$. The parameter $\theta$ belongs to a subset $\Theta$ of a Euclidean space $\mathscr{E}^q$ and is unkown. Let $g$ be a function over the parameter space having continuous first partial derivatives and taking values in $\mathscr{E}^r (r \leqq q)$. To test the hypothesis $g(\theta) = 0$ against the alternative $g(\theta) \neq 0$ using a sample of $n$ independent observations of $X$, one frequently uses the Neyman-Pearson generalized likelihood ratio test statistic $\lambda_n$. The limiting distribution of $-2\ln\lambda_n$ under the null hypothesis, as $n \rightarrow \infty$, was shown by Wilks (1938) to be chi-square with $r$ degrees of freedom (assuming regularity conditions). If $\{\theta_n\}$ is a sequence of alternatives converging to a point of the null hypothesis at the rate $n^{\frac{1}{2}}$, the limiting distribution is noncentral chi-square with noncentrality parameter equal to the limit of $n\lbrack g(\theta_n)\rbrack' \sum^{-1}_g (\theta_n)\lbrack g(\theta_n)\rbrack$, where $\sum_g(\theta)$ is the asymptotic covariance matrix of the quantity $n^{\frac{1}{2}}\lbrack g(\hat{\theta}) - g(\theta)\rbrack$ as $n \rightarrow \infty$ with $\theta$ fixed ($\hat{\theta}$ denoting the maximum-likelihood estimator of $\theta$ based on sample size $n$). This noncentral convergence was first proved by Wald (1943), along with a number of other results, on the basis of some rather severe uniformity conditions. Davidson and Lever (1970) have proved the result using more intuitive assumptions. Feder (1968) has obtained asymptotic noncentral chi-square for the case where both the hypothesis and alternative regions are cones in $\Theta$; this is essentially a generalization of $g(\theta) = 0$ versus $g(\theta) \neq 0$, since the hypothesis $g(\theta) = 0$ is locally equivalent to a hyperplane and $g(\theta) \neq 0$ to its complement. Despite the generality, Feder's assumptions are quite mild compared with Wald's. The result appears in Wald's paper as a special case of a more general statement entitled "Theorem IX." This theorem states that for $\theta \in \Theta$ and $-\infty < t < \infty$ the relationship \begin{equation*}\tag{1.1}P_\theta\lbrack -2 \ln \lambda_n < t\rbrack - P_\theta\lbrack K_n < t\rbrack \rightarrow 0\end{equation*} holds uniformly in $t$ and $\theta$, where $K_n$ has a noncentral chi-square distribution with $r$ degrees of freedom and noncentrality parameter equal to $n\lbrack g(\theta)\rbrack' \sum^{-1}_g (\theta)\lbrack g(\theta)\rbrack$. This formulation of Wald is too strong. It will be shown by counterexample that, if $\theta$ is held fixed while $n \rightarrow \infty$, relationship (1.1) fails to hold uniformly in $t$. The counterexample is that of testing the value of the mean of a normal distribution with unknown mean and variance. Wald's proof of Theorem IX treats two cases separately, case (i) where $\theta_n$ approaches the null hypothesis set at the rate $n^{-\frac{1}{2}}$ or faster, and case (ii) where it does not. The proof of (1.1) in case (i) requires convergence of $\theta_n$ at the rate $n^{-\frac{1}{2}}$ in order that the Taylor series expansion of the logarithm behave nicely. In case (ii) there is no reason at all to believe the distribution of $K_n$ to be a good approximation to that of $-2\ln\lambda_n$. From Wald's paper (page 480, line following (212)) one gets the impression that Wald felt that the statement of uniform convergence of (1.1) in case (ii) was trivial, since pointwise convergence is trivial (because both terms tend to zero for fixed $t$). But, since $K_n$ does not converge in distribution to a random variable in case (ii), there is really no reason why pointwise convergence should imply uniform convergence. In the same paper, Wald (1943) also described a test procedure based only on the unrestricted maximum-likelihood estimator $\hat{\theta}_n$. This procedure rejects for large values of the statistic $Q_n = n\lbrack g(\hat{\theta}_n)\rbrack' \sum^{-1}_g (\hat{\theta}_n)\lbrack g(\hat{\theta}_n)\rbrack.$ Wald claimed in his paper that (1.1) again holds uniformly in $t$ and $\theta$ if $-2\ln\lambda_n$ is replaced by $Q_n$. This claim too is false, in the stated generality, as the same counterexample will demonstrate. Keeping $\theta$ as a fixed alternative while $n \rightarrow \infty$ has the disadvantage that the limiting behavior of each of the quantities $-2\ln\lambda_n, Q_n$ and $K_n$ is degenerate in the sense that the probability mass moves out to infinity with increasing $n$. However, statement (1.1), uniform in $t$ for fixed $\theta$, has meaning here since both $-2\ln\lambda_n$ (or $Q_n$) and $K_n$ may be related to quantities with genuine limiting normal distributions which must be identical or at least very similar in order for (1.1) to be uniform in $t$. The precise result is embodied in a theorem presented in Section 2 of this paper. In Sections 3 and 4 we consider the case of $X$ normally distributed with mean $\mu$ and variance $\sigma^2$, where $-\infty < \mu < \infty, 0 < \sigma_1 < \sigma < \sigma_2$, and the hypothesis to be tested is $\mu = 0$. It is shown in Sections 3 and 4, respectively, that for this problem the relationships $P_\theta\lbrack Q_n < t\rbrack - P_\theta\lbrack K_n < t\rbrack \rightarrow 0$ and $P_\theta\lbrack -2\ln\lambda_n < t\rbrack - P_\theta\lbrack K_n < t\rbrack \rightarrow 0$ fail to be uniform in $t$ when $\theta = (\mu, \sigma)$ is fixed and satisfies $\mu \neq 0, \sigma_1^2 < \sigma^2 < \sigma_2^2 - \mu^2$. The space of values of $\sigma$ has been truncated in order to satisfy Wald's regularity conditions. In the following section boldface letters denote vectors and matrices. The law of the random vector $\mathbf{x}$ is denoted throughout by $\mathscr{L}(\mathbf{x})$. In particular, $\mathscr{N}(\mathbf{\mu}, \mathbf{\Sigma})$ refers to a normal law with mean vector $\mathbf{\mu}$ and covariance matrix $\mathbf{\Sigma}$. By $\mathscr{L}(\mathbf{x}_n) \rightarrow \mathscr{L}(\mathbf{y})$ or $\mathscr{L}(\mathbf{x}_n) \rightarrow \mathscr{N}(\mathbf{\mu}, \mathbf{\Sigma})$ is meant, respectively, that the law of $\mathbf{x}_n$ converges to the law of $\mathbf{y}$ or to the stated normal law, as $n \rightarrow \infty$. The definitions of the Mann-Wald symbols $O_p$ and $o_p$ may be found in Chernoff ((1956), Section 2), as may the statements of some basic results of large-sample theory which are used freely in the proof of the theorem.

This paper is concerned with the determination of whether two multivariate normal mean vectors differ by a multiplicative factor or by a multiplicative factor plus a displacement along a specified direction. These considerations arise, for example, in the evaluation of drugs, where two drugs affecting the same symptoms are considered equivalent if they can be made equal by a change in dosage. Maximum likelihood estimators and likelihood ratio tests for the relevant parameters are considered.

In this paper we compare the asymptotic efficiencies of several tests that are available for testing uniformity on the circle. Since the problem of testing goodness of fit on the circle can be reduced to testing uniformity by a simple probability transformation, these comparisons are applicable also to the goodness of fit situation. The alternatives to uniformity considered here are the familiar circular normal distributions (CND's) with density \begin{equation*}\tag{1.1}g(\alpha) = \lbrack 2\pi I_0(\kappa)\rbrack^{-1} \exp\lbrack\kappa \cos \alpha\rbrack,\quad - \pi \leqq \alpha < \pi.\end{equation*} $0 \leqq \kappa < \infty$ is a parameter of concentration, larger values of $\kappa$ corresponding to more concentration towards the mean direction zero, and $I_0(\kappa)$ is the Bessel function of purely imaginary argument. When $\kappa = 0 (1.1)$ is the uniform density, so the null hypothesis is $H_0: \kappa = 0$. The test compared here are (i) Ajne's test A (ii) Watson's test W (iii) Rayleigh's test R (iv) Ajne's test N (v) Kuiper's test V (iv) Spacings test U. In subsequent sections each of these tests is briefly described and its Bahadur efficiency [4], [5] is computed, using large deviation results. We compare the local slopes of the test statistics, i.e. the slopes in the neighborhood of the hypothesis. On the basis of these comparisons, we fine that limiting efficiencies of the first three tests viz., Ajne's test A, Watson's W and Rayleigh's test based on R, are identical, while the other tests have lower asymptotic efficiencies. Further conclusions are given in Section 7. Finally in Section 8 a simple inequality between the Ajne's N and Kuiper's V, whose asymptotic performances are identical, is noted.

An elegant characterization of optimal strategies for gambling problems was given by Dubins and Savage in the finitely additive setting of their book How to Gamble If You Must. An exposition of their ideas is given here in a measurable, countably additive framework. With the additional measurability assumptions, it becomes possible to treat a larger class of payoff functions. Also, necessary and sufficient conditions are given for a strategy to be $\epsilon$-optimal, a problem not considered by Dubins and Savage.

The concept of asymptotic normality takes on some new aspects when the dimensionality of the vector random variable under consideration is allowed to increase indefinitely. A necessary and sufficient condition for joint asymptotic normality in a new (strong) sense, in the case of independence, is given.

For a wide class of (dependent) random variables $X_1, X_2, \cdots, X_n$, a limit law is proved for the maximum, with suitable normalization, of $X_1, X_2, \cdots, X_n$. The results are more general in two aspects than the ones obtained earlier by several authors, namely, the stationary of the $X$'s is not assumed and secondly, the assumptions on the dependence of the $X$'s are weaker than those occurring in previous papers. A generalization of the method of inclusion and exclusion is one of the main tools.

Upper bounds are obtained for $|X(t)|/Q(t)$ as $t \rightarrow \infty$, where $X(t)$ is a continuous Gaussian process with $EX^2(t) \leqq Q^2(t), Q(t)$ non-decreasing. Our results are extensions of some work of Pickands (1967), Nisio (1967) and Orey (1971) to larger classes of Gaussian processes, i.e. fewer restrictions are imposed on the covariance functions. The results follow from Fernique's lemma (1964) and a recent lemma on the maximum of Gaussian sequences due to Landau, Shepp, Fernique and the author (see Marcus, Shepp (1971) for further references to this lemma).

Let $\{f(x, \omega); x \in R^n, \omega \in \Omega \}$ be an $n$ vector-valued stochastic process defined over a probability space $(\Omega, \mathscr{A}, \mu)$. Let $N(f \mid A, y)$ denote the number of elements in the set $A \cap f^{-1}(y)$, that is the number of distinct solutions of the system of equations $f(x, \omega) = y$ for $x, y \in R^n$. We develop expressions for $E\{N(f \mid A, y)\}$ and certain higher-order moments of $N(f \mid A, y)$ under regularity conditions.

Let $(X, \mathscr{U})$ be a pathwise connected and locally connected topological space with countable base, and $(\Theta, \mathscr{W})$ be a connected and locally connected continuous transformation group on $X$ which is Abelian. Let $\mathscr{A}$ be the smallest $\sigma$-field containing all open sets and $P \mid \mathscr{A}$ a probability measure such that $P(U) > 0$ for every open set $U \neq \varnothing$. For every $\vartheta \in \Theta$ let $P_\vartheta$ denote the probability measure generated by the transformation $\vartheta$, i.e. $P_\vartheta(A): = P(\vartheta^{-1} A), A \in \mathscr{A}$. Assume that $P_\vartheta$ admits a continuous density relative to $P$ for every $\vartheta \in \Theta$. Assume finally that for some sample size $n > 1$ there exists a real-valued, continuous statistic $T_n$ which is equivariant (i.e. $T_n(x_1, \cdots, x_n) = T_n(y_1, \cdots, y_n)$ implies $T_n(\vartheta x_1, \cdots, \vartheta x_n) = T_n(\vartheta y_1, \cdots, \vartheta y_n)$ for all $\vartheta \in \Theta$) and sufficient for $P_{\vartheta^n,} \vartheta \in \Theta$. Under these assumptions there exists a real-valued, continuous statistic $S$ on $X$ which is sufficient for $P_\vartheta, \vartheta \in \Theta$, such that the distribution of $S$ is either the location parameter family of normal distributions with variance 1 or a scale parameter family of gamma distributions. In a nutshell: Among the families of distributions which are generated by Abelian transformation groups, and which fulfill certain regularity conditions, the location parameter family of normal distributions and the scale parameter families of gamma distributions are essentially the only ones admitting for some sample size greater than one a sufficient statistic which is real valued, continuous and equivariant.

Let $P$ be a conservative and ergodic Markov operator on $L_\infty(X, \Sigma, m)$ (where $m(X) = 1$). It is proved that if for $A \in \Sigma$ with $m(A) > 0$ and $\mu \ll m$ a finite measure with $\mu(A) > 0 \lim_{n\rightarrow\infty} < \mu, P^{n+1}1_B >/<\mu, P^n 1_A>$ exists for every $B \subset A$, then $P$ has a $\sigma$-finite invariant measure $\lambda$ and there is a sequence $A_k \uparrow X$ with $A_0 = A$ such that for $0 \leqq f, g \in L_\infty(A_k) < \mu, P^n f>/<\mu, P^n g>\rightarrow < \lambda, f>/< \lambda, g>$. The result is used to study the convergence of $< \mu, P^n f>/< \eta, P^n g>$ for $\mu, \eta \ll m$, with applications to Harris processes and strong mixing point transformations. An analogous result for a positive contraction of $C(X)$ is given.

We study separable mean zero Gaussian processes $X(t)$ with correlation $\rho (t, s)$ for which $1 - \rho (t, s)$ is asymptotic to a regularly varying (at zero) function of $|t - s|$ with exponent $0 < \alpha \leqq 2$. For such processes, we obtain the asymptotic distribution of the maximum of $X(t)$. This result is used to obtain a result for $X(t)$ as $t \rightarrow \infty$ similar to the so-called law of the iterated logarithm.

This paper contains three results: (a) an exact measure function for the zero set of a real-valued asymmetric Cauchy process, (b) an upper bound for the Hausdorff dimension of the zero set of a real process with stationary independent increments, and (c) an exact lower function at infinity for the completely asymmetric Cauchy process.

Many discrete martingales with increments in $L_2$ can be normalized so that the resulting trajectory is distributed approximately like Brownian motion. This paper will find all such martingales, subject to a natural side condition. Two techniques of normalization are possible: The usual one involving the partial sums of conditional variances of the increments given the past, and the analogous method using the partial sums of squares of the increments. This result is applied to obtain a central limit theorem and an arc sin law for dependent random variables.

New $L_1$ and pointwise limit theorems are proved for the transition functions $P^n(x, E)$ of a Markov process with $\sigma$-finite invariant measure $\pi$ satisfying a recurrence condition. Also given are related results about the operators on functions and measures induced by these transition functions. The method depends upon the application of martingale theorems, and the principal restriction concerns the structure of a certain $\sigma$-field.

Consider a $GI/G/1$ queue in which $W_n$ is the waiting time of the $n$th customer, $W(t)$ is the virtual waiting time at time $t$, and $Q(t)$ is the number of customers in the system at time $t$. We let the extreme values of these processes be $W_n^\ast = \max \{W_j: 0 \leqq j \leqq n\}, W^\ast(t) = \sup \{W(s): 0 \leqq s \leqq t\}$, and $Q^\ast(t) = \sup \{Q(s): 0 \leqq s \leqq t\}$. The asymptotic behavior of the queue is determined by the traffic intensity $\rho$, the ratio of arrival rate to service rate. When $\rho < 1$ and the service time has an exponential tail, limit theorems are obtained for $W_n^\ast$ and $W^\ast(t)$; they grow like $\log n$ or $\log t$. When $\rho \geqq 1$, limit theorems are obtained for $W_n^\ast, W^\ast (t)$, and $Q^\ast(t)$; they grow like $n^{\frac{1}{2}}$ or $t^{\frac{1}{2}}$ if $\rho = 1$ and like $n$ or $t$ when $t > 1$. For the case $\rho < 1$, it is necessary to obtain the tail behavior of the maximum of a random walk with negative drift before it first enters the set $(-\infty, 0\rbrack$.

Countable state and action Markov decision processes are investigated, the objective being to maximize expected discounted reward. Well-known results of Maitra and Blackwell are generalized, their assumption of bounded rewards being replaced by weaker conditions, the most important of which is as follows. The expected reward to be received at time $n + 1$ minus the actual reward received at time $n$, viewed as a function of the state at time $n$, the action at time $n$ and the decision rule to be followed at time $n + 1$, can be bounded. It is shown that there exists an $\varepsilon$-optimal stationary policy for every $\varepsilon > 0$ and that there exists an optimal stationary policy in the finite action case.

Under certain conditions, it is shown that the invariant and almost-invariant $\sigma$-fields are equivalent if and only if the invariant $\sigma$-field is independent of an appropriate sufficient $\sigma$-field. This result helps unify work of Hall, Wijsman and Ghosh and of Pfanzagl, who dealt with the forward implication and work of Berk and Bickel, who treated the reverse implication. The conditions required are that the sufficient and invariant $\sigma$-fields be essentially disjoint and together generate the $\sigma$-field of the original data.

We consider some unresolved relationships among various notions of bivariate dependence. In particular we show that $P\lbrack T > t \mid S > s\rbrack \uparrow$ in $s$ (or alternately, $P\lbrack T \leqq t \mid S \leqq s\rbrack \downarrow$ in $s$) implies $S, T$ are associated, i.e. $\operatorname{Cov} \lbrack f(S, T), g(S, T)\rbrack \geqq 0$ for all non-decreasing $f$ and $g$.

This paper describes a way of investigating how the entries in a "raked" contingency table--one based on a given contingency table but having row and column totals equal to specified quantities in such a way as to minimize "discrimination information" between the two tables--vary as functions of the entries of the original table and the quantities to which the row and column totals of the "raked" table are constrained to be equal.

A convenient representation of the covariance matrix for a general $k$-level random, nested AOV model is obtained, as well as an expression for its inverse and determinant.

That the Kolmogorov-Smirnov statistics obey iterated logarithm laws is well known. For the purpose of developing nonparametric tests with power one it has become of interest to find accurate upper bounds for the probability that a sequence of Kolmogorov-Smirnov statistics ever exceeds a given boundary sequence. This paper is concerned with finding such probability bounds for a wide class of boundary sequences.

Let $q$ be the density function of the absolute value of a strictly stable random vector in $R^N, N$-dimensional Euclidean space. Asymptotic expressions for $q(r)$ for large $r$ and for small $r$ are found. The proofs use the Fourier inversion formula and contour integration. Bessel functions play a role occupied by the exponential and trigonometric functions when $N = 1$.

Pseudo-Poisson processes can be obtained from discrete time Markov processes by subordination. A continuous time analogue of a random walk is defined by $Y(t) = S\lbrack T(t)\rbrack$ where $S(n)$ is the partial sum of a sequence of independent identically distributed random variables and $T(t)$ a process with stationary independent increments, independent of $S(n)$ and taking values in the non-negative integers. It is then shown that $Y(t)$ is a compound Poisson process; furthermore the supremum of $Y(t)$ is Poisson-subordinated to the maximum of $S(n)$ if and only if $T(t)$ is a Poisson process.

It is well known that the first passage times for Brownian motion have stable laws with exponent $\frac{1}{2}$. It is shown here that first passage times for random walks have distributions in the domain of attraction of a stable law with exponent $\frac{1}{2}$.

Maruyama (1949) and Grenander (1950) derive necessary and sufficient conditions for stationary Gaussian processes on the real line or the integers to be metrically transitive. Their work is based on ergodic theorems for such processes. This paper studies conditions for metric transitivity for stationary Gaussian proceses for which there are no ergodic theorems. Instead the work is based on results on the absolute continuity of measures corresponding to random processes.

Let $(M_t, \mathscr{F}_t)_{t \geqq 0}$ be a continuous square integrable martingale and let $A_t$ be the natural increasing process in the Doob decomposition of $M_t^2$. Extending a result of Burgess Davis we show that there exist constants $C_1$ and $C_2$ such that $C_1 E\lbrack A_t^{\frac{1}{2}}\rbrack \leqq E\lbrack \sup_{s \leqq t} |M_s|\rbrack \leqq C_2 E\lbrack A_t^{\frac{1}{2}}\rbrack$ for all $t > 0$. Now if $A_\infty = \lim_{t\rightarrow \infty} A_t$, we find moment conditions on $A_\infty$ which relate to uniform integrability of $M_t$. In particular, $E\lbrack A_\infty^{\frac{1}{2}}\rbrack < \infty$ implies $M_t$ is uniformly integrable which implies $E\lbrack A_\infty^{1/\delta}\rbrack < \infty$ for all $\delta > 4$.

Let $X_{\alpha_1}(t)$ and $X_{\alpha_2}(t)$ be independent stable processes in $R_1$ of stable index $\alpha_1$ and $\alpha_2$ respectively, where $1 < \alpha_2 < \alpha_1 \leqq 2$. Let $X(t) \equiv (X_{\alpha_1} (t), X_{\alpha_2}(t))$ be a process in $R_2$ formed by allowing $X_{\alpha_1}$ to run on the horizontal axis and $X_{\alpha_2}$ on the vertical axis; $X(t)$ is called a process with stable components. The Blumenthal-Getoor indices of $X(t)$ satisfy $\alpha_2 = \beta" < \beta' = 1 + \alpha_2 - \alpha_2/\alpha_1 < \beta = \alpha_1$. Denote by $\dim E$ the Hausdorff dimension of $E$. It is shown that if $E = \lbrack 0, 1\rbrack$ and $F$ is any fixed Borel set for which $\dim F \leqq 1/\alpha_1$ then (with probability 1) we have $\dim X(E) = \beta' \dim E$ and $\dim X(F) = \beta \dim X(F)$. This shows that the results of Blumenthal and Getoor (1961) for the bounds on $\dim X(E)$ for arbitrary processes $X$ and fixed Borel sets $E$ are the best possible, and that their conjecture that $\dim X(E) = \dim X\lbrack 0, 1\rbrack \cdot \dim E$ is incorrect.