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- Question : 1 - Let g be a monotone function on a finite closed interval [a, b]. Show that a bounded function f defined on [a, b] is Riemann
- Question : 2 - Let Y?,h(t) be the random walk described in Section 1.2. Assume that h2 = o(?), i.e., h2/? ? 0 as ? ? 0. Show that X(t) = lim??0 Y?,h(t) exists and that X(t) ? 0
- Question : 3 - Let Y?,h(t) be the random walk described in Section 1.2. Show that for small ? and h we have Assume that ? ? 0, h ? 0, but h2/? ??. Then lim??0 Y?,h(t) does not exist. However, consider the following renormalization: Hence for this choice of ? and h, the limit lim??0 Y?,h(t) does not exist. However, the existence of the limit of the renormalization in Equation (1.2.5) indicates that the limit lim??0 Y?,h(t) exists in some generalized sense. This limit is called a white noise. It is informally regarded as the derivative ?B (t) of the Brownian motion B(t). Note that h = ?? satisfies the conditions h2/??? and h4/? ? 0 if and only if 14 < ? < 12

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