In this paper we introduce the concept of asymptotically almost negatively associated random variables in a Hilbert space and obtain the strong law of large numbers for a strictly stationary asymptotically almost negatively associated sequence of H-valued random variables with zero means and finite second moments. As an application we prove a strong law of large numbers for a linear process gen...

In preparation for the next post on the central limit theorem, it's worth recalling the fundamental results on convergence of the average of a sequence of random variables: the law of large numbers (both weak and strong), and its strengthening to non-IID sequences, the Birkhoff ergodic theorem. 1 Convergence of random variables First we need to recall the different ways in which a sequence of r...

Due to measurement uncertainty, often, instead of the actual values xi of the measured quantities, we only know the intervals xi = [x̃i − ∆i, x̃i + ∆i], where x̃i is the measured value and ∆i is the upper bound on the measurement error (provided, e.g., by the manufacturer of the measuring instrument). These intervals can be viewed as random intervals, i.e., as samples from the interval-valued rand...

Up to this point, we have only spoken of RVs as being single-dimensional objects. We will now turn to the case of vector-valued random variables, which are functions mapping a sample space into Rn. Thus, when we talk about the distribution or density of X = (X1, X2, . . . , Xn), we refer to n-dimensional real-valued functions. Alternatively, one could think of the individual elements Xi as dist...

Erratum to " The Clark-Ocone formula for vector valued random variables in abstract Wiener space " , Jour. In this paper we considered the extension of the Clark-Ocone formula for a random variable defined on an abstract Wiener space (W, H, µ) and taking values in a Banach space (denoted there either B or Y). The main result appears in Theorem 3.4. Unfortunately, as first pointed out to us by J...

Klaus Schmidt proved that if a strictly stationary sequence of (say) real-valued random variables is such that the family of distributions of its partial sums is tight, then that sequence is a \coboundary". Here Schmidt's result is extended to some (not necessarily stationary) sequences of random variables taking their values in a separable real Banach space.

We extend Fano’s inequality, which controls the average probability of (disjoint) events in terms of the average of some Kullback-Leibler divergences, to work with arbitrary [0, 1]–valued random variables. Our simple two-step methodology is general enough to cover the case of an arbitrary (possibly continuously infinite) family of distributions as well as [0, 1]–valued random variables not nece...

Two real-valued or vector-valued random variables X, Y are independent for probability measure P (written: X ⊥ Y [P ]) if for all sets A and B, P[X ∈ A, Y ∈ B] = P[X ∈ A] · P[Y ∈ B]. For jointly discrete or jointly continuous random variables this is equivalent to factoring of the joint probability mass function or probability density function, respectively. The variables X and Y are conditiona...