Encyclopédie des sciences mathématiques, tome IV-volume 2 by Jules Molk (ed)

By Jules Molk (ed)

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XN(tM)}, {tb ... ,tM} c I, is given and satisfies conditions (a) and (b) of consistency and symmetry stated earlier. 1 The Hilbert Space L2 (11) We are concerned with an important class of stochastic processes called second-order processes. The Hilbert space L2 (Q) occupies a central place in the discussion of properties associated with second-order stochastic processes. Hence, some preliminary remarks concerning the space L2 (Q) are in order. Let {Q, d, P} be a probability space and let X : Q ~ lB.

N}, the increments of Won [ti> ti+1), i = 1, 2, ... n, defined as are independent. , the trajectories of Ware continuous mappings of I into JR with probability one. In other words, at each fixed OJ E Q outside a set of probability zero, the function W (t, OJ) is continuous in tEl. d) EW(t) = 0, tEl. e) The increments {W(s) - W(t)} of Ware stationary in the sense that the expectation E {W(s) - W(t)}2 = IIW(s) - W(t)112 is a function of (s - t) only. ; 0 is an arbitrary constant. 79). A number of interesting properties of the standard Wiener-Levy process can be deduced from these conditions.

S. s. convergent if and only if it is a Cauchy sequence. s. 62) IE {XY}I ~ IIXIIIIYII and note the following useful properties. A) If Xn ~ X, then Ilxnll ~ IIXII and EXn ~ EX. 67), o Continuity of Inner Product. If Xn ~ X and Y m ~ Y as 00, then E{XnYm} ~ E{XY}. This result follows from Cauchy's inequality since B) n, m ~ + IE {Xn (Y - Ym)}1 IIx - XnllllYl1 + IIXnllllY - Ymll , IE {XY} - E {Xn Ym}1 ~ IE {(X - Xn) Y}I ~ which approaches zero as Ilxnll ~ Ilxll < 00 by virtue of (A). s. convergence.

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