If is the unique minimizer of then in (4.1.11) and (4.1.12) should be replaced by

Theorem 4.1.1 Assume A4.1.1, A4.1.2, and Conditions on hold.

Let be given by (4.1.9)-(4.1.12) (or (4.1.11)-(4.1.14)) with any initial value. Then

if and only if for each the random noise given by (4.1.10) can be decomposed into the sum of two terms in ways such that

with

and

where is given in Conditions on

Proof. We will apply Theorem 2.2.1 for sufficiency and Theorem 2.4.1 for necessity.

Let us first check Conditions A2.2.1–A2.2.4. Condition A2.2.1 is a part of A4.1.1. Condition A2.2.2 is automatically satisfied if we take noticing that in the presented case. Condition A2.2.4 is contained in A4.1.2. So, the key issue is to verify that given by (4.1.14) satisfies the requirements.

Let and be vector functions obtained from with some of its components replaced by zero:

It is clear that

For notational convenience, let denote a generic random vector such that

where is specified in (4.1.1), and may vary for different applications.

We express given by (4.1.14) in an appropriate form to be dealt with. We mainly use the local Lipschitz-continuity to treat the structural error (4.1.15) in

Rewrite the component of the structural error as follows

and for any express

where on the right-hand side of the equality all terms are cancelled except the first and the last terms, and in each difference of L, the arguments of L differ from each other only by one

Applying the Taylor’s expansion to (4.1.26) we derive

where

Similarly, we have

and

Define the following vectors:

Finally, putting (4.1.27)-(4.1.35) into (4.1.14) we obtain the following expression for

It is worth noting that each component of and is a martingale difference sequence, because both

and

For the sufficiency part we have to show that (2.2.2) is satisfied a.s. Let us show that (2.2.2) is satisfied by all components of and

For components of we have for any

since by (4.1.1), and as

Therefore, for any integer N

for any such that converges.

Thus, all sample paths of components of satisfy (2.2.2). Com- pletely the same situation takes place for the components of

By the convergence theorem for martingale difference sequences, we find that for any integer N

This is because is inde-

pendent of and is bounded by a constant uniformly with respect to by Lipschitz-continuity of Then the martingale convergence theorem applies since for some by A4.1.1.

Similar argument can be applied to components of Since for any integer N (4.1.38) holds outside an exceptional set with probability zero, there is an with such that for any

and

for all and N = 1,2, ….

Therefore, for all and any integer N

From (4.1.17) and (4.1.18) it follows that there exists such that and for each

and hence

Combining (4.1.41) and (4.1.42), we find for each

This means that for the algorithm (4.1.11)-(4.1.14), Condition A2.2.3 is satisfied on Thus by Theorem 2.2.1, on This proves the sufficiency part of the theorem.

Under the assumption a.s. it is clear that both and converge to zero a.s. and (4.1.39) and (4.1.40) turn to be

and

Then the necessity part of the theorem follows from Theorem 2.4.1. We show this. By Theorem 2.4.1, can be decomposed into two parts

and such that and Let us

denote by the component of a vector Define

and

From (4.1.43) and (4.1.36) it follows that

This together with (4.1.44) and (4.1.45) proves the necessity of the the- orem.

Theorem 2.4.1 gives necessary and sufficient condition on the obser- vation noise in order the KW algorithm with expanding truncations and randomized differences converges to the unique maximizer of a function

L. We now give some simple sufficient conditions on

Theorem 4.1.2 Assume A4.1.1 and A4.1.2 hold. Further, assume that

is independent of

and satisfies one of the following two conditions:

i) where is a random variable;

ii) Then

whre is given by (4.1.9)-(4.1.12).

Proof. It suffices to prove (4.1.16)-(4.1.18). Assume i) holds. Let

be given by

By definition, is independent of and so

where is an upper bound for

By the convergence theorem for martingale difference sequences, it follows that

Thus in (4.1.16) it can be assumed that

and and the conclusion of the theorem follows from Theorem 4.1.1.

Assume now ii) holds.

By the independence assumption it follows that for is inde- pendent of so that

Then, we have

It directly follows that

Again, it suffices to takes

We now extend the results to the case of multi-extremes. For this A 4.1.2 is replaced by A4.1.2’.

A4.1.2’ is locally Lipschitz continuous, L(J ) is nowhere

dense, where the set where L takes extremes, and

used in (4.1.11) is such that for some and

Theorem 4.1.3 Let be given by (4.1.9)-(4.1.12) with a given ini-

tial value Assume A 4.1.1 and A 4.1.2’ hold. Then

on an with if satisfies (4.1.16)- (4.1.18), or

satisfies conditions given in Theorem 4.1.2, where is a connected set contained in the closure of .

Proof. Condition A2.2.2 is implied by A4.1.2’ with and

A2.2.3 is satisfied as shown in Theorems 4.1.1 and 4.1.2. Then the conclusion of the theorem follows from Theorem 2.2.2.

Remark 4.1.3 In the multi-extreme case, the necessary conditions on

for convergence can also be obtained on the analogy of Theorem 2.4.2.

Remark 4.1.4 Conditions i) or ii) used in Theorem 4.1.2 are simple

indeed. However, in Theorem 4.1.2 is required to be independent of This may not be satisfied if the observation noise

is state-dependent. Taking into account that is the observation noise when observing at and we

see that depends on and if the observation noise is state-dependent. In this case, does depend on This violates the assumption about independence made in Theorem 4.1.2.

Consider the case where the observation noise may depend on loca- tions of measurement, i.e., in lieu of (4.1.3) and (4.1.4) consider

Introduce the following condition.

A4.1.3 Both and are measurable functions

and are martingale dif-

ference sequences for any and

for p specified in A4.1.1 with

where is a family of nondecreasing independent of both and

Theorem 4.1.4 Let be given by (4.1.9)–(4.1.12) with a given ini- tial value Assume A4.1.1, A4.1.2’, and A4.1.3 hold. Then

where is a connected subset of

Proof. Introduce the generated by and

i.e.,

It is clear that is measurable with respect to

and hence are Both

and are Ap-

proximating and by simple functions, it is seen that

Therefore, and are

martingale difference sequences, and

where

Hence, is a martingale difference sequence with

Noticing is bounded and as by (4.1.50) and (4.1.51) and the convergence theorem for martingale difference sequences we have, for any integer N > 0

This together with (4.1.37) with replaced by (4.1.39), and (4.1.40) verifies that expressed by (4.1.36) satisfies A2.2.3. Then the con- clusion of the theorem follows from Theorem 2.2.2.

Remark 4.1.5 If J consists of a singleton then Theorems 4.1.3 and

theorems ensure that converges to some point in J. However, the limit is not guaranteed to be a global minimizer of Depending on initial value, may converge to a local minimizer. We will return back to this issue in Section 4.3.

4.2.

Asymptotic Properties of KW Algorithm

We now present results on convergence rate and asymptotic normality of the KW algorithm with randomized differences.