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  2. Limit inferior and limit superior - Wikipedia

    en.wikipedia.org/wiki/Limit_inferior_and_limit...

    That is, xlim sup X n if and only if there exists a subsequence (X n k) of (X n) such that xX n k for all k. lim inf X n consists of elements of X which belong to X n for all except finitely many n (i.e., for cofinitely many n). That is, xlim inf X n if and only if there exists some m > 0 such that xX n for all n > m.

  3. Interchange of limiting operations - Wikipedia

    en.wikipedia.org/wiki/Interchange_of_limiting...

    Examples abound, one of the simplest being that for a double sequence a m,n: it is not necessarily the case that the operations of taking the limits as m → ∞ and as n → ∞ can be freely interchanged. [4] For example take a m,n = 2 m − n. in which taking the limit first with respect to n gives 0, and with respect to m gives ∞.

  4. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    A function is continuous at a limit point p of and in its domain if and only if f(p) is the (or, in the general case, a) limit of f(x) as x tends to p. There is another type of limit of a function, namely the sequential limit. Let f : X → Y be a mapping from a topological space X into a Hausdorff space Y, p ∈ X a limit point of X and L ∈ Y.

  5. Limit (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Limit_(mathematics)

    On the other hand, if X is the domain of a function f(x) and if the limit as n approaches infinity of f(x n) is L for every arbitrary sequence of points {x n} in Xx 0 which converges to x 0, then the limit of the function f(x) as x approaches x 0 is equal to L. [11] One such sequence would be {x 0 + 1/n}.

  6. Martingale central limit theorem - Wikipedia

    en.wikipedia.org/wiki/Martingale_central_limit...

    The martingale central limit theorem generalizes this result for random variables to martingales, which are stochastic processes where the change in the value of the process from time t to time t + 1 has expectation zero, even conditioned on previous outcomes.

  7. Kuratowski convergence - Wikipedia

    en.wikipedia.org/wiki/Kuratowski_convergence

    Let (,) be a metric space, where is a given set. For any point and any non-empty subset , define the distance between the point and the subset: (,):= (,),.For any sequence of subsets {} = of , the Kuratowski limit inferior (or lower closed limit) of as ; is ⁡:= {:,} = {: (,) =}; the Kuratowski limit superior (or upper closed limit) of as ; is ⁡:= {:,} = {: (,) =}; If the Kuratowski limits ...

  8. Doob's martingale convergence theorems - Wikipedia

    en.wikipedia.org/wiki/Doob's_martingale...

    The reason for the name is that if is an event in , then the theorem says that [] almost surely, i.e., the limit of the probabilities is 0 or 1. In plain language, if we are learning gradually all the information that determines the outcome of an event, then we will become gradually certain what the outcome will be.

  9. Fatigue limit - Wikipedia

    en.wikipedia.org/wiki/Fatigue_limit

    The fatigue limit or endurance limit is the stress level below which an infinite number of loading cycles can be applied to a material without causing fatigue failure. [1] Some metals such as ferrous alloys and titanium alloys have a distinct limit, [ 2 ] whereas others such as aluminium and copper do not and will eventually fail even from ...