8.7 Appendix

8.7.1 Derivations for Section 8.4.2

Derivation of (8.34)

Equation 8.132

Derivation of (8.38)

Equation 8.133

Derivation of (8.40)

Equation 8.134

Equation 8.135

( e _k is a K -dimensional vector with all-zero entries except for the k th entry, which is 1.)

8.7.2 Derivations for Section 8.4.3

Derivation of (8.56)

Equation 8.136

Derivation of (8.59)

Equation 8.137

Derivation of (8.61)

Equation 8.138

Equation 8.139

Derivation of (8.62)

Equation 8.140

Equation 8.141

Derivation of (8.63)

Equation 8.142

8.7.3 Proof of Proposition 8.1 in Section 8.5.2

Note that

Equation 8.143

The numerator in (8.143) is the target distribution, and the denominator is the sampling distribution from which Z _t was generated. Hence, for any measurable function h ( ·), we have

Equation 8.144

Finally note that both (8.90) and (8.91) are special cases of (8.144).

8.7.4 Proof of Proposition 8.2 in Section 8.5.3

In this section we verify the correctness of the residual resampling under a general setting. Let ( ) be a properly weighted sample with respect to p ( x _t Y _t ) “ without loss of generality, we assume that be the set of samples generated from the residual resampling scheme. The new set consists of copies of the sample for j = 1,..., m , and K _r = i.i.d. samples drawn from set with probability proportional to . The weights for the new samples are set to 1. Hence,

Equation 8.145

Furthermore,

Equation 8.146

Here we assume that Var { h ( x _t ) w _t } < . Hence, in probability.