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31
YflK 519.214
BecTHHK Cn6ry. Cep. 10, 2013, Bbm. 3
V. I. Norkin, R. J.-B. Wets
ON A STRONG GRAPHICAL LAW OF LARGE NUMBERS FOR RANDOM SEMICONTINUOUS MAPPINGS*)
1. Introduction. From the fundamental LLN (Law of Large Numbers) of Artstein and Vitale (1975) [1], Lyashenko (1979) [2], Artstein and Hart (1981) [3] one can immediately derive a strong pointwise LLN for osc random mappings; osc = outer semicontinuous, i. e., mappings with closed graphs. The (rich) potential applications to a variety of variational problems, however demand an a.s.-graphical LLN and not a pointwise one. More specifically, to be able to claim that the solutions of an inclusion, equivalently a generalized equation, of the type E{S(£, ■)} = S( ■) 3 0 can be approximated by the solutions of approximating inclusions Sv■) 3 0, a minimal condition is that almost surely the mappings Sv ■) converge graphically**) to S
This article is concerned with such a graphical LLN for (osc) random set-valued mappings, namely to provide conditions under which the graphs of their associated SAA-mappings, 'Sample Average Approximating' mappings, set-converge, i. e., in the Painleve-Kuratowski [4] sense, with probability one to the graph of the expectation mapping. Mostly, this study is a first step***) in validating the so-called SAA-method for a variety of variational problems such as stochastic variational inequalities, equilibrium problems in a stochastic environment (related to the GEI-model in economics), uncertainty quantification and so on, see [4, §5.F], for example.
As mentioned earlier, the first LLN [1, 2] were obtained for integrably bounded random sets (in Rm), later generalized [3] to simply 'integrable' random sets, i.e., admitting an integrable selection, but not necessarily bounded; a.s.-convergence has to be understood as set-convergence to the closure of the convex hull of the Aumann's [5, 6] expectation of the random set. These results were extended to infinite dimensions, dependent and fuzzy random sets, cf. reviews by Taylor and Inoue (1996) [7], Molchanov (2005) [8], Li and Yang (2010) [9]. The extension from random sets to random mappings, i. e., depending on parameters,
Norkin Vladimir Ivanovich — doctor of physical and mathematical sciences, leading scientific researcher, 03187, Kiev, V. M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine; e-mail: [email protected].
Wets Roger J.-B. — distinguished research professor of mathematics, Ph. D. engineering sciences, CA 95616-5270, Department of Mathematics, University of California, Davis, USA; e-mail: [email protected].
*) The paper is based on the report at the International conference 'Constructive Nonsmooth Analysis and Related Topics' (CSNA-2012), June 18-23, 2012, Euler International Mathematical Institute, St. Petersburg, Russia. The work of Vladimir Norkin was supported by a Fulbright Fellowship while staying at the Department of Mathematics of the University of California, Davis (2011), and by Russian-Ukrainian grant $40.1/016 (2011-2012) of the Ukrainian and Russian State Funds for Fundamental Research. For Roger Wets, this material is based upon work supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office under grant number W911NF1010246.
**) Other convergence notions, like pointwise, for example, either don't yield the convergence of the solutions or the more demanding convergence notions, such as uniform or continuous convergence, fail to be applicable except when resorting to supplementary conditions that often restrict inappropriately the range of applicability.
***) Only a first step, because we restrict our attention mostly, but not exclusively, to compact-valued mappings. We do this, in part, to make the presentation more accessible but also to elucidate the relationship with the limited existing literature.
© V. I. Norkin, R. J.-B. Wets, 2013
is a qualitatively new problem because one has to select a new topology to analyze the convergence of not necessarily continuous mappings.
There are only a couple of papers that attempt to deal with this problem: Shapiro and Xu (2007) [10] and Teran (2008) [11] studied LLN of bounded-valued, integrably bounded, random set-valued mappings with respect to the uniform norm. Shapiro and Xu [10] proved the uniform convergence of SAA mappings to a certain fattened expectation mapping, but a genuine uniform LLN in their setting only holds for the case when the expectation mapping is continuous. Teran [11] treats sets as elements of the so-called convex combination metric space, equips set-valued mappings with the uniform metric and then applies the LLN due to Teran and Molchanov (2006) [12]. He derives a uniform LLN under the crucial, but restrictive, assumption that the essential range of the random mapping is separable with respect to this uniform metric which renders it only applicable in quite restrictive settings.
We proceed as follows: we first prove, under the existence of a uniform integrable bound, that the graphs of SAA-mappings a.s.-converge to the graph of the expectation mapping; this convergence is equivalent to the convergence of graphs with respect to the Pompeiu-Hausdorff distance. Next, we show that the pseudo-uniform LLN by Shapiro and Xu [10], is in fact equivalent to the graphical LLN when restricting ourselves to their framework*) . As already indicated earlier, applications of the graphical LLN are mostly aimed at obtaining approximating solutions of stochastic generalized equations, stochastic variational inequalities and stochastic optimization problems with equilibrium constraints all involving, usually, unbounded mappings except when specific, if not artificial, restrictions are introduced, see, e. g., Shapiro and Xu (2008) [17], Xu and Meng (2007) [18], Ralph and Xu (2011) [19].
Section 2 introduces notation, concepts and some basic facts concerning set-valued mappings. Section 3 reviews, for reference purposes and later use, some known results about the law of large numbers for random sets and mappings. In Section 4, we prove the graphical LLN for random mappings uniformly bounded by an integrable function and bring to the fore the limitations of the pseudo-uniform LLN of Shapiro and Xu [10].
2. Notation, definitions, and preliminaries. Our terminology and notation is pretty consistent with that of [4].
2.1. Set-valued mappings. Let X be a closed subset of the complete separable metric space H (e. g., Rn or more generally, a separable Banach space) with distance dist( •, •), ]Rm be m-dimensional Euclidean space with inner product (•, •) and Euclidean norm | • |. Denote by cpct-sets(]Rm) the hyperspace of compact subsets of ]Rm and cl-sets(]Rm) the hyperspace of closed subsets of Rm. Introduce the distance from a point x to a set A and the excess of the set A on B as
da(x) = dist(x,A) = inf dist(x',x), e(A, B) = sup inf d(a,b)
aeAbeB
*) However, it should be noted that these results are not indiscriminately applicable to unbounded random mappings, e. g., to random cone-valued mappings, although some easy, simple, extensions are possible; for example when the osc-mapping random is the sum of a compact-valued osc mapping (with a uniform integrable bound) and a constant-valued, possibly unbounded, osc-mapping. A graphical LLN for an important, but a very specific class of unbounded random mappings, namely for epigraphical random mappings, was proved under a variety of assumption by Attouch and Wets (1990) [13], King and Wets (1991) [14], Artstein and Wets (1995) [15] and Hess (1996) [16]. These latter results state that epigraphs of SAA-lsc (lower semicontinuous) functions a.s. converge to the epigraph of the expectation functional.
and doo (A, B) the Pompeiu-Hausdorff distance between the sets A and B,
do(A, B) = max {e(A, B), e(B, A)} .
Denote by Bp(x) C Rm the ball centered at x with radius p, IB the (closed) unit ball and \\A\\ =supaeA ¡4
For a set A c Rm define its support (= Minkowski's) functional as aA(u) = supaeA (a, u). For sets {Si C Rm, i = 1,...,v} define their (Minkowski's) average by
Sv = v-1 Si = s s = si, si G Si >
i=i I i=i J
and for mappings ^ : H ^ Rm, i = l,...,v} define their (Minkowski's) pointwise averaged mapping Sv: for x G X, x ^ Sv (x)
Sv(x) = Si(x) = < s = si, si G Si(x), i = 1,...,v > .
i=i I i=i J
Definition 2.1 (set convergence, [4, Definition 4.1]). Define the inner and outer limits of a sequence of sets Sv C H,
Liminf v Sv = {x G H\3 xv G Sv ,xv ^ x} ,
Limsupv Sv = {x G H \ 3 {vk} CN,xk G Sv and xk ^ x} .
A sequence of sets Sv converges to a set S = Limv Sv if
Liminfv Sv = Limsupv Sv = S.
Definition 2.2 (osc and e-osc). A mapping S : X ^ cl-sets(Rm) is called outer-semicontinuous (osc) at x relative to X if for any p > 0 and any e > 0 there exists a neighborhood B$^ep,)(x) = {x' G H \ dist(x',x) ^ S(e,p)} of x such that for all
x' G BS(s,p)(x) n X
S(x') n Bp C S(x)+ B£.
Furthermore, it's e-osc at x relative to X if for any e > 0 there exists a 6 > 0 such that for all x' G IBs(x) n X, e(S(x), S(x')) ^ e or equivalently, S(x') C S(x) + eB. Finally, S is osc or e-osc on X if it's osc or e-osc at every x G X. Definition 2.3 (graphical limits of mappings, [4, Definition 5.32]). The mappings Sv : X ^ cl-sets(Rm) defined on a subset X C H are said to converge graphically to a mapping S relative to X, denoted Sv S or S = g-Limv Sv, if graphs of Sv, as sets, converge to the graph of S in the product space X x Rm, i. e.
gph Sv = {(x, s) G X x Rm\s G Sv(x)} ^ gphS.
Note, that the limiting mapping S = g-Limv Sv always has a closed graph and, consequently is osc; also, a constant sequence of osc mappings Sv = S graphically converges to itself.
For the product space X x Rm define the distance between z' = (x',y') and z = (x,y) by Dist(z',z) = dist(x',x) + \y' — y\ with the corresponding Pompeiu-Hausdorff distance between sets. When gph Sv and gph S are compact subsets of a bounded region in this
product space, then graph-convergence is equivalent to their convergence with respect to the Pompeiu-Hausdorff distance, but that's definitely not the case in general.
at all x e X, where unions U{xv_xy are taken over all sequences {xv ^ x}c X. As already mentioned earlier, outer semicontinuity of the limit mapping S is an immediate consequence of graphical convergence (see [4, Definition 5.32]).
2.2. Random sets and random set-valued mappings. Let X be a closed subset of (H, dist), a complete separable metric space, BX be the Borel a-algebra of subsets of X, (S, £~,P) be a P-complete probability space. One refers to convergence with probability one in this space, also as almost sure (a.s.-convergence); for more about random sets and measurable mappings, refer to [4, Ch. 14; 8].
Definition 2.4 (random sets). A mapping S : S ^ cl-sets(Rm) is a random set if it is measurable, i.e. for any open subset O C IRm one has
Definition 2.5 (random mappings). A set-valued mapping S : S x X ^ cl-sets(J?m) is called a random mapping, if its graph, gph S, is a random set in the space X x Rm equipped with Borel a-algebra BX x BRm.
Definition 2.6 (iid random sets and mappings). Random sets {Si : S ^ cl-sets(IRm), i = 1, 2,...} are independent identically distributed (iid) with respect to measure P if (a) P{S-1(Bi),i e I} = []iei P{S-1(B i)} for all Bi e BRm, i e I, and all finite subsets I C {1, 2,...} and (b) P{S- 1(B)} = P{s-1(B)} for all B e BRm and all i,j.
Random mappings {Si : S x X ^ cl-sets(IRm),i = 1,2,...} are iid, if their graphs {gphSi} are iid random sets in X x Rm.
Let us remind that Aumann's expectation/integral [5, 6] of a random set is defined as a collection of expectations of all P-summable selections of this set. A random set-valued mapping at every fixed point is a random set and thus can be integrated pointwise.
3. LLNs for random sets and mappings. We review, for reference purposes, the law of large numbers (LLN) for random sets by Artstein and Hart (1981) [3], the epigraphical LLN of Attouch and Wets (1990) [13], and a pseudo-uniform LLN for random mappings due to Shapiro and Xu (2007) [10].
Theorem 3.7 (LLN: unbounded closed random sets, [3]). Let {S, Si,i = 1,...} be iid closed random sets in IRn with IES = %. Then, for the averaged sets Sv = V=1 S\,
one has,
where cl con denotes a closure of the convex hull.
For compact sets, this LLN goes back to Artstein and Vitale (1975) [1].
Theorem 3.8 (an epigraphical LLN, [13]). Suppose H is a separable Banach space, (S, En,P) is a complete probability space and {fi : S x H ^ (-m, is a sequence of
pairwise iid random lsc functions, bounded below P-almost sure by a polynomial minorant, fi(Z,x) > -aQ\\x - xo\\p - ai(£) with p e [1, m), xo e H, a0 e R+ and a1 integrable. Then,
*) Refer to the proof of this proposition to observe that these inclusions remain valid when x is the subset of a Polish space.
{xv —>x}
{xv —>x}
S-1(O) = {£ G S I n O = ^e Es.
Lim^ Sv = cl con IES a.s.,
for P-almost all Ef^, ■ ) = e-lim - V ■ ).
v V z—'
i=1
The epigraphical limit, e-lim, means set-convergence of the epigraphs of corresponding functions. Epi-convergence, in particular, yields ([4, Proposition 7.2])
liminfj, > Ef
for all xv ^ x G H a.s.
Theorem 3.9 (pseudo-uniform LLN, compact-valued mappings, [10], see also [18, Lemma 3.2]). Assume
(a) the metric space (X, d) is compact;
(b) {Si(£,x),i = 1,...} is an iid sequence of realizations of the random mapping S : S x X ^ cpct-sets(jRm) and Sv(£, x) = v-1 YH=1 Si(£, x);
(c) there exists a P-integrable function k : S ^ IR1 such that
IIS(£,x)|| < k(£) V(£,x) G S x X;
(d) for P-almost all £ the mapping S(£, • ) is e-osc.
Then, the expectation mapping ES = E{con S(£, • )} is well-defined and the compact-valued mapping ES is itself e-osc. For any p > 0, one has a double 'one-sided uniform' convergence for P-almost all namely
limv
sup e(Sv(£,x),ESp(x))
xex
0 = limv
sup e(ES(x),S;(£,x))
xex
(2)
with the fattened-up mappings
ESp(x) = UyeBp{x)ES(y), SV(z, x) = UyeBp{x)S1vy).
Teran [11] by relying on an abstract LLN due to Teran and Molchanov [12] (for convex combination metric spaces), obtained a strong uniform LLN for a random mapping S(Z,x), however under the essential assumption of separability of 'a' range of S, namely, for some measurable subset S' of S of P-measure 1, the set rge S = U^h' S(Z, •) is separable with respect to the dist~-metric in the space of set-valued mappings. This metric is defined as follows: for mappings S1 and S2, dist~(S1,S2) = supxeX d00(S1(x),S2(x)). Unfortunately, this assumption is not fulfilled in very simple and natural situations as confirmed by the example below.
Example 3.10 (dist~ range separability fails). Define a set-valued mapping
( 0, 0 < x<Z,
s(e,x) = I [0,i], x = z,
[ 1, z <x < 1,
where Z is a random variable uniformly distributed on [0,1], x G [0,1]. Then any essential range of S(Z,x) (a subset of [0,1]2) is not separable with respect to the uniform dist~-metric, so Teran's (2008) [11] uniform LLN would not be applicable in this case.
So, essentially this leaves us with only one 'genuine' LLN by Shapiro and Xu [10] when p = 0, in other words, when ES is continuous
*) Let's observe that Teran and Molchanov (2006) [12] obtain a somewhat related result but this time for a different notion of expectation of random sets, i.e., not of the Aumann's type.
4. A strong graphical LLN for random mappings: Uniformly bounded case.
The aim of this section is to establish a graphical LLN for random set-valued mappings and show that the pseudo-uniform law of large numbers for uniformly bounded (by an integrable function) random set-valued mappings due to Shapiro and Xu [10] is, for an appropriately restricted class of random mappings, a graphical LLN. This fact allows to substitute sample average approximations v-1 ^V=1 Si(£, ■) = Sv(£, ■) 3 0 for an inclusion 1ES(£, ■) 3 0, where {Si,i = 1,...,v} are independent identically distributed versions of a random osc mapping S(£, ■). Suppose IE{S(£,x)} = cl con IE{S(£, x)}, as is the case for convex-, compact-valued bounded mappings, then a.s.-graphical convergence (LLN) Sv(£, ■) —— ES(£, ■) implies by [4, Theorem 5.37] convergence of the solutions of the associated inclusions; again the proof of the referenced theorem applies without modifications to the case when H is a Polish space.
Theorem 4.11 (a.s.-graphical-LLN for compact-valued random mappings). Assume
(a) X is a closed subset of a separable Banach space H, BH is the Borel a-algebra and (S, En, P) is P-complete;
(b) mappings S(£, x) = S0(£,x), Si(£,x) : S x X — cpct-sets(Rm), i = 0,1,... are nonempty-valued, x BH-jointly measurable with respect to (£,x) and osc in x G X for P-almost all £ G S, i. e. the graphs gphSi(■) are closed random sets in X x Rm;
(c) the random graphs gph Si are iid with the same distribution as gph S;
(d) there is an integrable function k(£) such that
sup{ |s| | s G Si(£,x)} < k(£) V i, V (£,x) G S x X.
Let Sv(£, x) = v-1 J2V=1 S\(£,x) and S, the mapping whose graph, gphS, is the graph of IE con S(£, ■). Then S is osc and Sv —— Sa.s. on X.
Proof. Let's prove graphical convergence Sv —— S a.s. on X by checking criterion (1). First let's prove the left inclusion by relying on Theorem 3.2. The right inclusion will be proved in the subsequent Lemma 4.2.
Let D be a countable dense subset of Rm. For d G Rm, define the support functions
a(£,x; d) = sup (y,d), ai(£, x; d) = sup (y,d),
yes(£,x) yeSi(t,x)
1 v
a"(€,x;d) = sup (y, d) = - V} cr¿(£, x; d), yesv(£,x) v i=1
a(x; d) = sup {(y,d) | y G lE{conS( ■, x)}}.
Let us check applicability of the LLN of Theorem 3.2 to the random functions
—ai(£, x; d), x G X;
âi(£,x; d) = ^ x G H \ X.
Under boundedness (d), the osc mappings S(£, • ), Si(£, ■ ) are e-osc for any fixed £ and hence their support functions a(£, ■ ; d), ai(£, ■ ; d) are upper semicontinuous in x G X [21, § 3.2, Proposition 2] and ài(£, x; d) are lsc with respect to x G H. For a fixed d, the functions â ( ■, ■ ; d) are £3 x BH-measurable, indeed
{(£, x) G S x H 1 âi(£, x; d) > c} =
= S x (H \ X) U {(£,x) G S x X I Si(£,x) n Bd = 0} G £3 xBH
by joint-measurability of Si, where Bd = {(x, y) G X x Rm | (y, d) > c}g Bh x BRm, c G R.
Then by [22, Lemma VII.1, Theorem III.30] functions ai (Z, •; d) are normal integrands and also random lsc functions [13] (cf. [4, Definition 14.27, Corollary 14.34]). Note that by (d), a(Z,x; d) ^ —\d\fc(Z) for all x G H.
Let's now verify that {ai(Z, •; d)} are iid. First show that the random mappings {x ^
sf(Z,x)},
Sd(z x) i {—(s, d) \ s G Si(Z x)}, x G X,
i (Z, ) \ x G H \ X,
are iid, then the epigraphs
{epiai(Z, •; d)=gphSd(Z, • ) + (0 x R+)}
would be iid by [13, Lemma 1.2], where the zero vector 0 g H. Indeed for any B1 G BH, bounded B2 G BR one has
P{gph Sd(Z, •) n B1 x B2 = 0} = P{gph Si (Z, •) n (B1 n X) x Bid = 0},
where Bd = {y G Rm \ —(y,d)GB2}G BRm. Hence, the mappings {Sd(Z, •)} are identically distributed. For any Bi1 G BH, bounded Bi2 G BR, i G I C{0,1,...},
P{gphSd(Z, •) n Bi1 x Bi2 = 0,i G I} = P{gphSi(Z, •) n (Bi1 n X) x Bid2 = 0,i G I} = = n P{gph Si(Z, •) n (Bi1 n X) x Bd2 = 0} = n P{gph Sd(Z, •) n Bi1 x Bi2 = 0},
iei iei
where Bd2 = {y G Rm \ — (y,d) G Bi2} G BRm, hence the mappings {Sd(Z, •)} are independent. Now applying Theorem 3.2 to {ai( •, •; d)}, one obtains
e-lim ^ ¿^(Z', '; d) j = Ea(Z, •; d),
for all Z' G S\Sd, P{Sd} = 0. This, in particular, means that for any sequence {X 3 xv ^ x} when Z' G S \ Sd,
1 v
limsupj, — ^^ <jj(Z', xu\d) = limsup^ o"I/(Z', xu\d) ^ Ea(Z, x; d). v i=1
This is also true for all d G D when Z' G S' = S \ UdeDSd, i. e., with probability P{S'} = 1. Denote by R(Z, x; d) = sup{(s, d) \ s G Limsupv Sv(Z, x)}. Since for {X 3 xv ^ x}, x G X for all d G D,
limsupv R(Z', xv; d) ^ limsupv av(Z', xv; d) ^ lEa(Z, x; d) = a(x; d).
Taking into account cl con ES(Z, x) = IE cl con S(Z, x) = IE con S(Z, x) by [8, Theorem 1.17(iii)] and the fact that S(Z,x) is compact, this allows us to conclude
Limsupv Sv(Z', xv) C cl con ES(Z, x) = E con S(Z, x) = S(x),
and hence the left inclusion (1) holds jointly for all x G X with probability one. For the converse inclusion in (1), see the next lemma. □
The following lemma proves the converse inclusion (1) for the sample average mappings S"(£, x) = Yli= i x) for all x G X a.s. Note that in this lemma we do not assume boundedness of the random mappings. The proof exploits essentially the pointwise LLN of Theorem 3.1.
Lemma 4.2 (Liminf inclusion). Let's assume:
(a) X is a closed subset of a complete separable metric space and (S, Eh,P) is P-complete;
(b) mappings {S (£, x) = S0(£,x), Sj,(£,x) | S x X ^ IRm, i = 0,1,...}, are nonempty closed-valued, Eh xBRm -measurable in (£,x), i. e., the graphs gph Si(£, •) are random closed sets in X x Rm;
(c) the random graphs {gphS, (gph Si, i = 1,...) C X x IRm} are iid;
(d) ES(£,x) = 0 for all x e X.
Let Sv(£,x) = V=1 Si(£,x) and S be the mapping whose graph, gphS, is the closure of
the graph of con IE{S(£, •)}, gph S = clgphcon EE{S(£, •)}. Then, for P-almost all £ e S,
cl con E{S(£,x)} C S(x) C U{xv_x}LiminfvSv(£,xv).
Proof. Obviously, cl con IE{S(£, x)} C S(x). Let's prove the second inclusion. Choose a countable dense subset G in gph S (any subset of a separable metric space, in our case gphS C X x IRm, is also separable [20, Section 16.7]) and denote by X' its (countable) projection on X. For each x' e X' by the pointwise law of large numbers, Theorem 3.1, one has Sv (£, x') ^ cl con ES(£, x') a.s. Since X' is countable, this is true for all x e X' jointly a.s., i.e., for all £ e S' for some S' with P{S'} = 1.
Now, fix £' e S' and z = (x,y) e gph S. We need to show that limv Dist(z, gphSv(£', •)) = 0. Suppose, to the contrary, for some e > 0 and some subsequence {vk}, Dist(z, gphSVk) > e. By definition of G, there exists z'(e) = (x',y') e G with x' e X' such that Dist(z,z') ^ e/3. From the set convergence of Sv(£',x') ^ cl con ES(£,x'), it follows [4, Proposition 5.33] that
cl con ES(£, x') C Liminfv Sv(£', x') C Liminf Svk (£', x').
Hence for the given e and y' e cl con ES(£, x') one can find vk and yvy e Svy (£', x') such that | y' - yvy | < e/3. Then, for this subsequence vk',
Dist(z, gph SVk' (£', •)) < Dist(z, z') + Dist(z', gph SVk' (£', •)) <
< Dist(z, z') + Dist(z', (x', yVk')) < Dist(z, z') + \y' - yVk' | < 2e/3
which contradicts the assumption that Dist(z, gph Svy (£', •)) > e.
Thus, limv Dist(z, gphSv(£', •)) =0, i. e., there is a sequence zv e gphSv(£', •) such that zv —> z G gph S. □
The next proposition shows that the "uniformity" statements in (2) are in fact equivalent to the graphical convergence of the involved mappings, Sv S.
Proposition 4.3 (uniform characterization of graph-convergence). Graphical convergence Sv S of compact-valued mappings to an osc mapping S : X ^ cpct-sets(Rm) on a compact set X C Rn is equivalent to
limsup e(Sv(x),Sp(x)) = 0 = lim sup e(S(x),Svp (x)) Vp > 0, (3)
v xex v xex
where
SP(x) = uyeBp(x) S(У), SV(x) = {JyeBp(x)Sv (y).
Proof. Let Sv S with S osc on X and let's prove (3). By [4, Exercise 5.34] for any r > 0 and e > 0 for all x G X n Br and v sufficiently large,
Sv(x) n Br C S(Be(x)) + Be = Se(x) + Be,
S(x) n Br C Sv(Be(x)) + Be = Sv(x) + Be.
Fix any p > 0, set dJ = supxeX ||x|| < and M = sup{\y\ \ y G S(x),x G X} < since X is compact and S is osc, compact-valued. For any e < p, r > max{dJ, M + p}, and for x G X the preceding inclusions become
Sv(x) C Se(x) + Be, S(x) C Sv(x) + Be.
From this, it follows
e(Sv(x), Sp(x)) < e(Sv(x), Se(x)) < e,
Jp
V Jp
e(S(x), Sv(x)) < e(S(x), SV (x)) < e.
Thus, for any e and sufficiently large v, one has supxeX e(Sv(x),Sp(x)) < e and supxeX e(S(x), S"p (x)) < e which is what we set out to prove.
Let's now concern ourselves with the converse, namely that (3) implies graphical convergence SS on X. We begin by showing that the first identity of (3) implies e(gph Sv, gph S) — 0. For any x,
e((x,Sv(x)), gphS) = sup Dist((x,y), gphS) <
yesv (x)
^ inf ( dist(x, xX)+ sup dist(y, S(x')))
x'ex ^ yesv(x) '
= inf (dist(x, x') + e(SV(x),S(x')) .
x' EX
The inequality supxeX e(Sv (x), Sp (x)) < e means that for each x there exists xp such that dist(x,xp) ^ p and e(Sv(x),S(xp)) ^ e, so
e((x,SV(x)), gph S) < dist(x, xp) +e(SV(x),S(xp) < p + e,
and consequently e(gph SV, gphS) < p + e. Since p and e can be arbitrary small, it means that e(gph SV, gph S) — 0 as v — to.
Similarly, from supxEX e(S(x), SV(x)) — 0, one obtains e(gph S, gph SV) — 0 as v — <x>. Hence, the Pompeiu-Hausdorff distance d(gph S, gph SV) — 0 as v — <x>. Under outer semicontinuity of S, recall it means that gph S is closed, this is equivalent to SV —— S and completes the proof. □
Example 4.4 (SAA of Clarke's subdifferential). In [10], Shapiro and Xu consider sample average approximations of the expectation of the Clarke's subdifferential IE{0f(£, •)} of random Lipschitz functions f (£, •) which is interpreted to mean that for all x — f (£, x) is Lipschitz continuous.
Detail. Convergence of the SAA-versions is 'proved' under a reguarity assumption and the requirement that the Lipschitz constant be integrable. However, as seen from Theorem 4.1 to validate this approximation one needs the joint (£,x)-measurability of 0f(£,x); a proof of this property can be found in [23, Lemma 4]. □
References
1. Artstein Z., Vitale R. A. A strong law of large numbers for random compact sets // Ann. Probab. 1975. Vol. 3. P. 879-882.
2. Lyashenko N. N. Limit theorems for sums of independent, compact, random subsets of Euclidean space // J. of Soviet Mathematics. 1982. Vol. 20. P. 2187-2196 (Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR. 1979. Vol. 85. P. 113-128).
3. Artstein Z., Hart S. Law of large numbers for random sets and allocation processes // Math. Oper. Res. 1981. Vol. 6. P. 485-492.
4. Rockafellar R. T., Wets R. J.-B. Variational Analysis. Grundlehren der Mathematischen Wissenschaften. Berlin: Springer-Verlag, 1998 (3rd Printing in 2009). 734 p.
5. Richter H. Verallgemeinerung eines in der Statistik benötigten Satzes der Masstheorie // Math. Ann. 1963. Vol. 150. P. 85-90 & 440-441.
6. Aumann R. J. Integrals of set-valued functions //J. Math. Anal. Appl. 1965. Vol. 12. P. 1-12.
7. Taylor R. L. Inoue H. Laws of large numbers for random sets // Random Sets / eds.: J. Goutsias e. a. New York: Springer-Verlag, 1997. P. 347-366.
8. Molchanov I. Theory of Random Sets. London: Springer-Verlag, 2005. 488 p.
9. Li S., Yang W. Capacities, set-valued random variables and laws of large numbers for capacities // Integrated Uncertainty Management and Applications. AISC 68 / eds.: V.-N. Huynh e. a. Berlin: Springer, 2010. P. 127-138.
10. Shapiro A., Xu H. Uniform law of large numbers for set-valued mappings and subdifferentials of random functions // J. Math. Anal. Appl. 2007. Vol. 325. P. 1390-1399.
11. Teran P. On a uniform law of large numbers for random sets and sub-differentials of random functions // Statist. Probab. Lett. 2008. Vol. 78. P. 42-49.
12. Teran P., Molchanov I. The law of large numbers in a metric space with a convex combination operation // J. Theoret. Probab. 2006. Vol. 19. P. 875-898.
13. Attouch H., Wets R. J.-B. Epigraphical processes: Law of large numbers for random lsc functions // Seminaire d'Analyse Convexe, Montpellier. 1990. P. 13.1-13.29.
14. King A., Wets R. Epi-consistency of convex stochastic programs // Stochastics and Stochastics Reports. 1990. Vol. 34. P. 83-92.
15. Artstein Z., Wets R. Consistency of minimizers and the SLLN for stochastic programs // J. Convex Anal. 1995. Vol. 2. P. 1-17.
16. Hess C. Epi-convergence of sequences of normal integrands and strong consistency of the maximum likelihood estimator // Ann. Probab. 1996. Vol. 24. P. 1298-1315.
17. Shapiro A., Xu H. Stochastic mathematical programs with equilibrium constraints, modeling and sample average approximation // Optimization. 2008. Vol. 57. P. 395-418.
18. Xu H., Meng F. Convergence analysis of sample average approximation methods for a class of stochastic mathematical programs with equality constraints // Math. Oper. Res. 2007. Vol. 32. P. 648-668.
19. Ralph D., Xu H. Convergence of stationary points of sample average two-stage stochastic programs: a generalized equation approach // Math. Oper. Res. 2011. Vol. 36. P. 568-592.
20. Kolmogorov A. N., Fomin S. V. Introductory Real Analysis. New York: Dover Publications, 1975. 403 p.
21. Aubin J.-P., Ekeland I. Applied Nonlinear Analysis. New York: Wiley, 1984. 518 p.
22. Castaing C., Va,la,d,ier M. Convex Analysis and Measurable Multifunctions. Lecture Notes in Mathematics 580. Berlin: Springer-Verlag, 1977. 278 p.
23. Norkin V. I. Stochastic Lipschitz functions // Cybernetics. 1986. Vol. 22. P. 226-233. (Translated from Kibernetika. 1986. No. 2. P. 66-71 & 76).
Статья рекомендована к печати проф. В. Ф. Демьяновым. Статья поступила в редакцию 21 марта 2013 г.
