RANDOM FIELDS AND THEIR GEOMETRY

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Contents 1 Random fields 1 1.1 Random fields and excursion sets . . . . . . . . . . . . . . . 1 1.2 Gaussian fields . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 The Brownian family of processes . . . . . . . . . . . . . . . 6 1.4 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.1 Stochastic integration . . . . . . . . . . . . . . . . . 13 1.4.2 Moving averages . . . . . . . . . . . . . . . . . . . . 16 1.4.3 Spectral representations on R N . . . . . . . . . . . . 19 1.4.4 Spectral moments . . . . . . . . . . . . . . . . . . . 24 1.4.5 Constant variance . . . . . . . . . . . . . . . . . . . 27 1.4.6 Isotropy . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.4.7 Stationarity over groups . . . . . . . . . . . . . . . . 32 1.5 Non-Gaussian fields . . . . . . . . . . . . . . . . . . . . . . 34 2 Gaussian fields 39 2.1 Boundedness and continuity . . . . . . . . . . . . . . . . . . 40 2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2.1 Fields on R N . . . . . . . . . . . . . . . . . . . . . . 49 2.2.2 Differentiability on R N . . . . . . . . . . . . . . . . . 52 2.2.3 Generalised fields . . . . . . . . . . . . . . . . . . . . 54 2.2.4 Set indexed processes . . . . . . . . . . . . . . . . . 61 2.2.5 Non-Gaussian processes . . . . . . . . . . . . . . . . 66 2.3 Borell-TIS inequality . . . . . . . . . . . . . . . . . . . . . . 67 2.4 Comparison inequalities . . . . . . . . . . . . . . . . . . . . 74 This is page iii Printer: Opaque this
Page 1: RANDOM FIELDS AND THEIR GEOMETRY Ro
Page 5 and 6: 0 Contents 5 Non-Gaussian geometry
Page 7 and 8: 2 1. Random fields Definition 1.1.2
Page 9 and 10: 4 1. Random fields with mj = E{Xj}
Page 11 and 12: 6 1. Random fields space T is what
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Page 15 and 16: 10 1. Random fields FIGURE 1.3.2. C
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Page 19 and 20: 14 1. Random fields will be to esta
Page 21 and 22: 16 1. Random fields 1.4.2 Moving av
Page 23 and 24: 18 1. Random fields independent of
Page 25 and 26: 20 1. Random fields Theorem 1.4.3 (
Page 27 and 28: 22 1. Random fields Since indicator
Page 29 and 30: 24 1. Random fields FIGURE 1.4.1. T
Page 31 and 32: 26 1. Random fields The correspondi
Page 33 and 34: 28 1. Random fields Proof. Isotropy
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Page 37 and 38: 32 1. Random fields where C is now
Page 39 and 40: 34 1. Random fields where the Wml a
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Page 45 and 46: 40 2. Gaussian fields Their diversi
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Page 49 and 50: 44 2. Gaussian fields Corollary 2.1
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48 2. Gaussian fields From (2.1.12)
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50 2. Gaussian fields both is assur
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52 2. Gaussian fields If sN > tN th
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54 2. Gaussian fields 2.2.3 General
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56 2. Gaussian fields Two things ar
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58 2. Gaussian fields We have to sh
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60 2. Gaussian fields Before leavin
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62 2. Gaussian fields done in full
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64 2. Gaussian fields with the boun
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66 2. Gaussian fields 2.2.5 Non-Gau
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68 2. Gaussian fields Theorem 2.3.1
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70 2. Gaussian fields But (2.3.8) n
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72 2. Gaussian fields On the other
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74 2. Gaussian fields so that �
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76 2. Gaussian fields Proof. We hav
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78 2. Gaussian fields where the ξn
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80 2. Gaussian fields which exists
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84 2. Gaussian fields Since �∞
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86 2. Gaussian fields Differentiati
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88 2. Gaussian fields Furthermore,
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90 2. Gaussian fields all of which
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94 2. Gaussian fields It is easy to
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96 3. Geometry We shall not make ma
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98 3. Geometry 1, . . . , m. The co
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100 3. Geometry Then if we write N(
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102 3. Geometry where ϕ(A ∩ E x
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104 3. Geometry Since the main purp
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106 3. Geometry The inverse mapping
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108 3. Geometry f(0, x) = u or f(1,
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110 3. Geometry guage, will be exte
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112 3. Geometry All told, this can
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114 3. Geometry In other words, FIG
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116 3. Geometry where Ej is any j-d
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118 3. Geometry cube) a level of ge
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120 3. Geometry borhood of x we hav
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122 3. Geometry manifold (M, g). Th
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124 3. Geometry Let T (V ) = ⊕∞
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126 3. Geometry We start with Λ n,
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128 3. Geometry 3.5.3 Tensor bundle
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130 3. Geometry where D 1 ([0, 1];
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132 3. Geometry Euclidean inner pro
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134 3. Geometry Recall that, for an
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136 3. Geometry We now return to th
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138 3. Geometry where sign(f ∗ )
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140 3. Geometry the curvature tenso
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142 3. Geometry Finally, we note on
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144 3. Geometry where R X ijkl = n
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146 3. Geometry If we now define Γ
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148 3. Geometry 3.7.1 Piecewise smo
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150 3. Geometry Note that the union
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152 3. Geometry The time has come f
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154 3. Geometry a notion of ‘loca
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156 3. Geometry In this decompositi
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158 3. Geometry could be used, via
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160 3. Geometry for 0 ≤ j ≤ N
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162 3. Geometry Consistent with thi
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164 3. Geometry normal cones. Conve
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166 3. Geometry it can be decompose
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168 3. Geometry (i) Points t ∈ (I
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170 3. Geometry
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172 4. Gaussian random geometry pro
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174 4. Gaussian random geometry Whi
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176 4. Gaussian random geometry (c)
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178 4. Gaussian random geometry The
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180 4. Gaussian random geometry and
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182 4. Gaussian random geometry Let
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184 4. Gaussian random geometry Tak
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186 4. Gaussian random geometry and
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188 4. Gaussian random geometry (d)
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190 4. Gaussian random geometry As
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192 4. Gaussian random geometry The
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194 4. Gaussian random geometry Pro
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196 4. Gaussian random geometry the
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198 4. Gaussian random geometry Lem
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200 4. Gaussian random geometry (4.
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202 4. Gaussian random geometry to
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204 4. Gaussian random geometry We
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206 4. Gaussian random geometry FIG
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208 4. Gaussian random geometry Goi
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210 4. Gaussian random geometry (e)
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214 4. Gaussian random geometry It
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218 4. Gaussian random geometry 4.8
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220 4. Gaussian random geometry by
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222 4. Gaussian random geometry whe
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224 4. Gaussian random geometry whe
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226 4. Gaussian random geometry Thu
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228 4. Gaussian random geometry Con
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230 4. Gaussian random geometry als
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232 4. Gaussian random geometry we
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234 4. Gaussian random geometry G,
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236 4. Gaussian random geometry Cle
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282 References [9] J-M. Azaïs and
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284 References [42] H. Hadwiger. No
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286 References [74] B. O’Neill. E
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288 References [103] E. Wong. Stoch
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RANDOM FIELDS AND THEIR GEOMETRY

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