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This definition is technically called Q-convergence, short for quotient-convergence, and the rates and orders are called rates and orders of Q-convergence when that technical specificity is needed. § R-convergence , below, is an appropriate alternative when this limit does not exist.
The definition of convergence in distribution may be extended from random vectors to more general random elements in arbitrary metric spaces, and even to the “random variables” which are not measurable — a situation which occurs for example in the study of empirical processes. This is the “weak convergence of laws without laws being ...
with = a small change of in the j direction, and () = the corresponding rate of change in the probability distribution. Since relative entropy has an absolute minimum 0 for P = Q {\displaystyle P=Q} , i.e. θ = θ 0 {\displaystyle \theta =\theta _{0}} , it changes only to second order in the small parameters Δ θ j {\displaystyle \Delta \theta ...
Convergence in total variation is stronger than weak convergence. An important example of a log-concave density is a function constant inside a given convex body and vanishing outside; it corresponds to the uniform distribution on the convex body, which explains the term "central limit theorem for convex bodies".
The rate of convergence is linear, except for r = 3, when it is dramatically slow, less than linear (see Bifurcation memory). When the parameter 2 < r < 3, except for the initial values 0 and 1, the fixed point = / is the same as when 1 < r ≤ 2. However, in this case the convergence is not monotonically.
Another discrete-time process that may be derived from a continuous-time Markov chain is a δ-skeleton—the (discrete-time) Markov chain formed by observing X(t) at intervals of δ units of time. The random variables X (0), X (δ), X (2δ), ... give the sequence of states visited by the δ-skeleton.
This expression asserts the pointwise convergence of the empirical distribution function to the true cumulative distribution function. There is a stronger result, called the Glivenko–Cantelli theorem, which states that the convergence in fact happens uniformly over t: [5]
On the right-hand side, the first term converges to zero as n → ∞ for any fixed δ, by the definition of convergence in probability of the sequence {X n}. The second term converges to zero as δ → 0, since the set B δ shrinks to an empty set. And the last term is identically equal to zero by assumption of the theorem.