A system of transition probabilities is non-null if for all n ∈ Z and all ω − ∞ n − 1 ∈ A − ∞ n − 1 P [ ω ( n ) | ω − ∞ n − 1 ] > 0. Following [50], a chain with complete connection μ is a Gibbs measure consistent with the system of transition probabilities p ( n, ⋅ ) if this system is continuous and non-null.
If and if is continuous, then.
Then is exponentially convex if and only if is continuous and (1.10).
then there exist and such that for any if and only if is continuous at.
If f is continuous, this yields the ((M, N -convexity of f; that is, f bigl(M(x, y; 1-lambda,lambda) bigr) leq N -convexityf(y); 1-lambda, lambda bigr) fof every (x, y in I) and every (lambdain[0, 1]).
Notice that there is no need of increasing requirement on g in Corollary 12 if F is continuous and this property is used only when X has MCB property.
No continuity conditions are required for this result, but if is continuous in the real variable for each fixed, then is -linear, and if is continuous at a single point of, then is also continuous.
Moreover, if is measurable or if is continuous in for each fixed, then for all and.
This property obviously holds if η is continuous since η attains its maximum (less than 1) on each compact [ t, t + δ ( t ) ]. Definition 22 A sequence { x n } is said to be asymptotically T-regular with respect to f if lim n → ∞ d ( f x n, T x n ) = 0.
Secondly if it was assumed that this variable is continuous, then the analysis methods would assume a normal distribution for this measure.
