a new approach to series of independent random variables

Socicdad de Estad(st ica c Invest igacidn Opcrativa

Test (2001) Vot. 10, No. 2, pp. 405 418

A new approach to series of independent random

variables Ricardo V~lez*

Departamento de Estadistica e tuvestigaci6n Operativa U~dversidad Nacional de Educacidn a Distancia, Spain

Abstract

The aim of this paper is to provide an alternative approach to the classical theory of sums of independerd random variables. It shows that the Kohnogorov inequalities may be avoided in the proof of the three series theorem, and the equivalence lemma follows fl'om a very simple argument. The main idea is to relate the a.s. convergence of a series to the fact that their paths remain bounded.

K e y Words; Equivalence lemma, independent random variables, series, three series theorem. AMS subjec t classification: 60F15

1 I n t r o d u c t i o n

Let {Xj} be a sequence of independent random variables, defined in some

probabil i ty space ( ~ , 5 , P ) , and S.~ ~-~=1Xj.

The s tudy of the convergence propert ies of sequences {S~} is one of the

main topics in any advanced tex tbook on Probabi l i ty Theory, once these

propert ies have been established by Kolmogorov, L~vy, Doob, LoSve, etc.

The usual approach begins with the celebrated Kohnogorov inequalities,

as a technical requirement to deduce the "Three series theorem". This

result is then considered as a master key in order to obtain tile principal

consequences concerning tile convergence of series.

On the other hand, the "equivalence lemma" (which provides equiva-

lence between ahnost sure convergence and convergence in law) is habi tnal ly

established by means of characteristic functions.

'"Correspondence to: Ricardo V61ez, Departamento de Estadfstica e I.O., UNED, Madrid, Spain. Emaih r.velez(g, ccia.uned.es

Received: July 2001; Accepted: September 2001

406 R. Vdlez

In the following sections I will propose a different approach, which may give a greater insight into the behaviour of the sequence {Sn} .

Section 2 considers the symmetr ic case, in which the reflection principle allows us to prove that {Sn} converges a.s. if and only if it is almost surely bounded or bounded in probability. From this, the equivalence lemma is readily obtained. Furthermore, the "Two series criterion" can be then established without using the Kohnogorov inequalities.

In Section 3 the previous results are extended to the general case. As shown in Lemma 2, the key idea is to relate tile convergence properties of {S~} with those of the symmetrized series {Sn}, by means of a well known device. In this ww, tile equivalence lemma and the "Three series theorem" are again easily proved.

Section 4 gives a simple example where different behaviours of {St,} and {S~} can be observed.

2 T h e s y m m e t r i c c a s e

Tile case in which all tile terms [Xj} have a symmetr ic distribution (i.e.

_h~. d -A~) is tile simplest one, not only because each &, has a symmetr ic distribution, but mainly due to the fact that the sign of each term Xj may be changed without disturbing the distribution of {Sn}, whenever the change is made independently of the size of Xj .

For instance, the signs may be reversed from some fixed time r , in order to obtain the symmetr ic path:

S'~ = ~ &* if n _< r [ S~ - ( Sn - Sr ) if n > r.

Thus { & } and {S '} will have the same distribution: all the paths { & ] can follow, can also be followed by {s'} with the same probability (see Figure 1).

More precisely, the t ime r m w be aW stopping time, depending on the t ra jectory until t ime r. This means that {r = k} depends only on the values of X1, . . �9 X~ and is independent of X~+I, Xk+2, �9 �9 �9 In this case the distribution of {Sn} remains unchanged if the signs of the terms X~+l, X~+2, . . . are reversed. This m w be stated as:

Series of independent random variables 407"

. . . . . . . . . . . . . . . 4

Sn

I T 7~

Figure 1: The symmeh'ic path.

L e m m a 2.1. (Reflection Principle) For any stopping t ime r , the sequence

A first application of the reflection principle is the following well known proof of the Ldvy inequalities.

T h e o r e m 2.1. (L4vy inequalities) For any a > 0

( ] P { m a x S k > a } < 2P{S.~ > a} and

J

~}. t '~-<~ J

Proof. In {max~<.n Sk _> a, S,~ < a}, the time r first t ime wi th Sk >_ a takes place before n, and ST _> a; thus S: n 2S t - S.~ >_ a. Therefore

P ~maxSk > a,S.~ < a ; < P{S,~' > a} = P{S~ > a} [ t_<~ J

and

< 2 P { & > a}.

The right hand side is P{l&l k a) since & is symmetric. Pm'thermore

408 R. Vdlez

since min~<.~ S~ nmx~<~( S~) d _ = _ = maxk_<~ S~. []

Tile reflection principle may be applied iteratively, at first with respect to tile stopping time vl, then with respect to a later stopping time v.+, and so on, and tile distr ibution of tile t ransformed sequences always remains unchanged. In this way, if {:5.} is an increasing sequence of stopping times, the terms X j between (vl, v2], (va, v4], and so on, change their signs, giving rise to the alternate sequence with value, for n E [~-r, ~u

C o r o l l a r y 2.1. For any increasing sequence of stopping t imes {~:r}- the

sequence {S~[} has the same distribution as {S~}.

It is well known that (as any sequence of numbers) tile partial sums of a numerical sequence can fluctuate and remain bounded failing to converge. This behavior is not possible if the terms are chosen independently at random and with symmetric distribution.

T h e o r e m 2,2, {S~} is bounded a.s. i f and only i f it is a.s. convergent.

Pro@ Any bounded sequence failing to converge fluctuates up and down some fixed interval [a, hi, which can be assumed to have rational endpoints. Thus, fix a < b E Q and let A~,5 be tile event

A~,b -- {cJ: for all n E N there exists nl , n2 > n with S.n~ (w) > b

and S.~ (w) < a}

of all the trajectories in which there are infinitely many values greater than b and infinitely many values lower than a (see Figure 2). Let us consider

~-l - - first t ime wi th ,gk > b, r'zr = first t ime, after ~'2r-], with ,g~ < a,

r'~r+l - first t ime, after r'~r, wi th S~ > b

which is an increasing sequence of stopping times. Thus the associated sequence {S~} has the same distr ibution as {S.;~}.

In Aa, b all tile stopping times ~=r are finite and, since 'g~2~ ,g~, ._~ < a b

and,.%2,+~ ,g~.2,.>b a, t h e n S . ~ > b + ( r 1)(b a) forT~<n<T~+l . That means A~,b C {S~ is unbounded} and, as {S.~} remains bounded a.s., we have P(A~,b) 0. Hence P(U~<bcQ A~,b) 0 and {S~} converges a.s. []

Series of independent random variables 409

.• I I I I I

1 7-2 ~-3 7-4 ~-5 7-6

Figure 2: A trajectoiy in A~,6.

Tile idea of the event A~,b is due to Doob and is well known in martingale theory, where the expected number of upcrossings may be bounded. Here the reflection principle allows a simpler argument.

Now note that tile events

{S~ converges} C [S~ is bounded}

are tail events not depending on any concrete term Xj , but only on the asymptot ic behavior of the sequence {Xj}. According to the Kohnogorov Zero-One law (Chung 1974, pp. 254), each of these sets has probabil i ty 0 or 1, and the preceding result shows that they have the same probability. Thus, there are only two kinds of series of independent and symmetric random variables:

> Series bounded with probabil i ty one, which converge a.s. to a finite limit S.

> Series unbounded a.s., satisfying P f l i m sup & = +co} = P [ l i m inf S~ = - e c } - 1 (since {Sn} and {-Sr~} are identically distributed).

{S~} is bounded a.s. if and only i fP [ex i s t s K > 0 with supn I&l < K} = 1 and, since the event {sup~ I&l < K} is increasing with K, this nlay be s tated as

lira P { s u p I&l < K} - 1 or lira P{snp IS,,I _> K} - O. K ) c ~ n K ~ o ~ n

Moreover, the sequence {S..} is said to be bounded in probabil i ty if and only if

s-pP[Is l > K} 0 as K -4

410 R. Vdlez

or, in other words, if the sequence of distributions of {&~} is tight. The Ldvy inequality shows that there is no difference between bo th properties (for sums of independent and symmetric random variables) and we can therefore prove.

T h e o r e m 2.3. {Sn} is bounded in probability if and only if it is a.s. con- vcrgcnt.

Pro@ If for any c > 0 we can find K > 0 such ttmt, for all n,

P{I&I > K} < it is P{ maxl&l > K} < k < n

Thus, as n --+ co, we get P{sup k I&l > _< []

This result can be found in Skorohod (1986, pp. 31) where the proof is based on the three series theorem, obtained fl'om tile Kohnogorov inequalities. An immediate consequence not s tated there is the following.

T h e o r e m 2.4. (Equivalence lemma) {Sr~} converges in dist~ibution if and only if it converges a.s.

In f~ct, it is a well known elementary fact that any sequence of dis- t r ibutions converging in law is tight. Consequently, if S~ converges in distr ibution it is bounded in probability.

The preceding proof nmst be compared with Chung (1974, pp. 347) or Chow (1978, pp. 274).

If ,~bj(t) is the (real) ctmracteristic function of Xj , Sn tins ~n( t ) - n t 1]j=l ~b( ) as characteristic function. Thus, by means of the LSvy conti-

mfity theorem (Chung 1974, pp. 160), the convergence of tPn(t ) to some characteristic function is a necessary and sufficient condition for the a.s. convergence of S~. However, there exists a well known equivalent condition easier to check.

As a first step, let us assume that each t e r m X j has a finite variance cr~.

2 ~ ~r2 be the variance of S~. and let s n = E j 1 2

C~5 T h e o r e m 2.5. I f ~ j=~ (r~j < c~z., then {S~} converges a.s.

2 I f ~'~ f f oo and the terms X j arc bounded by some constants 7j such

that maxj<r~ 7j/S'n ~ O, then {S~} is unbounded with probability one.


Proof. If s2r~ grows to a finite limit i f , the Tchebychev inequality gives

P{IS,~I _> K} _< ~.~,/K ~ l ,e,,ce s,pP{IS,~l _> K} _< J / K ~ > 0 '1%

as K --+ oo. There%re {Sn} is bounded in probability and converges a.s.

2 When s~ f f oo and I~;I _< ~j with maxj<,~Gj/s,~ --+ O, it follows from the Liapunov Central Limit Theorenl (Chnng 1974, pp. 200) that Sn/~'r~ converges in law to the A/(0, 1) distribution. But the a.s. convergence of {5'~} to a finite limit would give that ,g,~/s,~ converges a.s. to 0, so certainly {S~} is unbounded a.s. []

Tile use of tile Liapunov theorem is suggested in Love (1977, pp. 262). It can be avoided by means of the following consequence (Fristedt-Gray 1997, pp. 204) of the Cauchy-Schwarz inequality: for any ), G (0, 1),

__~ E [ s 4 j l / 2 p { s 2 > /~82}1/2- I- )kS. 2.

Since ~[S 4] E r~) 1 ~[ X413 + 3 Ei#jn : E [ X ~ ] E [ X ] ] _< 72s. n2 + 3s.n ,4 taking ), = 1/2, for all n such that s,~/2 > K s, we wil l have

4 4 2 S .n S r~

P{IS,~I _> K} > P{S~ _> ~/2} > 4e[,s'~-~ -> 4(G2~2,~ + 3~4~)

and the last te rm converges to 1/12. Hence {Sn} is not bounded in probability.

The last result can be easily stated as a necessary and sufficient condition for the a.s. convergence of {S~}, which does not need assumptions about the variance of each term.

T h e o r e m 2.6. {S~} converges a.s. if and only if

(# Ej~-: P{IXjl > ~} ~nd

converge for some c > 0 (and then both series converge for any c > 0).

412 g. V41ez

Proof. If S.~ is a.s. convergent, for any g > 0, the inequality [Xj[ > g can hold only for finitely many terms X j . By the Borel-Cantelli lemma, this is equivalent to the convergence of (1).

Then tile series Sn ~ = 1 Xj and S(n s) ~jn_ 1 Xj I{[xo[<_s } have the same behavior. So, S~ converges a.s. if and only if (1) is convergent and

S~ e) converges a.s. But, as the terms of the last series are bounded by s, it converges a.s. if and only if (2) is convergent. []

C o r o l l a r y 2.2. I f E[[S~I v] is a bounded sequence, for some p > O, then { & } ~o~wrg~ ~.s. Morrow,', { & } ~onw~g~ ~.~. 'if ~j%m E[lXjlPJ ] < oo for" some sequence pj E (0, 2].

Proof. If H is a bound for Ell&F], the Markov ineqnality gives

P { I & I > K} < K-----7-- < K--7 - -~ 0 when K ~ ec.

Hence {S~} is bounded in probability and converges a.s. to a limit S (in this case E[IS~I ~] ---> E[ISI ~] and S~ --+ S in Lr, for each r < p; see Chung 1974, pp. 95-97).

For any pj E (0, 2], we have

P{IXj l > 1} < m[lxjlVJl{ixjl>l}] < EUIXjI~J]

Hence, from ~~jo~ ] < o% the convergence of the two series of Theorem 2.6 follows, proving that {S~} is a.s. convergent (whenpj = p < 1,

it is [S~[ p < ( E ~ I [Nil) p < E}* 1 [Nil p, and EUIS~[ p] is bounded). []

3 T h e genera l case

When the symmetry hypothesis of the terms {X j} is dropped, it remains true that the ~'random part" of {&~} cannot be oscillating in a finite interval. However, one can add a constant 6'.n to each partial sum Sn (or cj - Cj - 6'0._1 to each term Xa. ) in order to force an arbitrary fluctuating character of S.~ + C.~. Fortunately, the non-random component of {S.~} can be easily renloved. In fact, we can associate with {S~,} a symnletric series whose behavior informs us about the character of {S.~}. The method for doing


this is frequently used in tile theory of stochastic process with independent increments (Oihman and Skorohod 1975, pp. 260).

Let ( ~ ' , 7 , P ' ) and {X5} be copies of tile probabil i ty space (~,br, P ) , and tile te rms {Xj} defined up on it. Both sequences can be considered as defined in tile p roduc t space (~ x ~' , ,7" | Y , P x P ' ) wi th {2~} and {X5} independent and identically: dis tr ibuted. If S;~ ~ j = l Xr {S~,} and {S.'n} are also independent and identically distr ibuted. Moreover, tile series

j = l

has independent and symmetr ic r andom terms.

L e m m a 3.1. {,g~} converges a.s. if and only if there eecists a sequence of

Proof. If {S.~ - C,,} converges with (P • P ' ) probabil i ty one, so does {S~, - T h e n & - ( & - - (SI - co. erges a . s .

On tile other hand, assume tha t {,g~} converges in an (co, J ) set with (P • P ' ) probabil i ty one. Then, except for w' in a set N ' with P' (N ' ) - O, tile event

A~., { w : ,~n(ca,w') ,g.n(ca) ' ' } = = Sn(w ) converges

satisfies P(A~,) - 1. This implies, tha t almost all tile trajectories {Cn - S'~,(w')} of the twin series ( that are constants, not depending on ca) are able to make {S.~ C.~} convergent with (P) probabil i ty one. []

It must be noted tha t tile sequence {C.~} is not at all unique, not only because almost any t ra jectory gives such a sequence, but also because, if {C~,} is any sequence such tha t C = lim(C~ 6'.[~) exists, • 6'~, converges to S + C whenever ,g~ C.~ converges to S.

The existence of sequences of constants making {S.~} convergent has been considered by Doob, and tile preceding result, a l though not so stated, can be found in Doob (1953, Ch. III), where an a lwws suitble choice of tile constants C n is given. Tile "essential convergence lemma '~ in Love (1977, pp. 262) has exactly tile s ta tement of our previous result, but the proof is based on tile three series theorem. Here we will use it to extend tile propert ies established in tile synnnetr ic case.

414 R. Vdlez

T h e o r e m 3.1. {S~} is bounded in probability ,if and only ,if there exists a bounded sequence of constants {C~} such that {Sn - C,+} converges a.s.

Proof. {,~} is bounded in probability when {S~} (and therefore {S~}) is bounded in probability. This follows since, as I S~,I < I S ,~ I+IS ' I , we have

supP{IS,, I _> K} _< supP{ISn I _> K / 2 } + supP{IS~I _> K / 2 } : 0 f# /z ?~

when K --+ ec. Therefore {,~n} converges a.s. and a sequence of constants {C~} exists such that {&+ C~} converges a.s.

If a subsequence C~,,. diverges to +0% the subsequence S~,. must diverge to +oc a.s., and this is not compatible with tile fact that S~, is bounded in probability (for any K > 0, it is limv P{inf~,>v IS~,,.,I _> K} 1 and limv P{S~,. > K} = 1). In the same way, there is no subsequence C~,. diverging to 0% and C~ remains bounded.

Conversely, if {S~, - C~} converges a.s., it is bounded with probability one and, if {6'i+} is bounded, {&+} is bom, ded a.s. []

Accordingly bounded in probability and bounded a.s. are equivalent properties for {S~,}. This last result can be found in Skorohod (1986, pp. 32), but its proof is again based on tile three series theorem. We can fornmlate an immediate consequence.

T h e o r e m 3.2. (Equivalence lemma) {S~,} converges in distribution if and only if it converges a.s.

Proof. {Sn} is bounded in probability when it is convergent in distribution, and bounded constants C~ exist such that S~ 6'~ converges a.s. to a finite limit S. Two subsequences C n, and Cn,, cannot have different limits C ~ and C", since this would give S~, - S~, - C~, + C~, +'*S S + C' and

S~,, = S~,, C~,, + C~,, E:f4 S + C" and, as S + C' d S + C", we nmst have C' - C". Hence tile sequence C~+ has a linfit C and Sn converges a.s. (an additional simplification of the argument is due to an a n o w m o u s referee). []

The usual proofs of this result are certainly more involved. For instance, tile proof in Love (1977, pp. 263) again uses tile three series criterion.


If ,~Sj(t) is tile characteristic flmction of Xj, the characteristic flmction of & is %,(t) [I}~ ~ %bj(t) and ]@r,(t)] 2 tile characteristic funct ion of S,~. Thus, {S,} converges a.s. if and only if %,(t) converges to a characteristic function, while ]~P.n (t)12 converges to a characteristic funct ion if and only if {S~ C~} converges a.s. for some constant sequence C~. Both s ta tements are habi tual ly useless, but Theorem 2.6 (with each Xj replaced by ~:j) gives a simpler criterion for the a.s. convergence of {S,~}.

In agreement with Loire , the series {&,} will be called essentially co,~- vm'gmlt if {St,} is a.s. convergent. Otherwise, {S.n} is te rmed essentially diyergent; this type of series are a.s. unbounded and, according to the Kol- mogorov Zero-One law, the vector

[P{liminf& -~},P{lin, sup& +~}]

can take tile values [0, 1], [1, O] or [1, 1]. Similarly, tile tail events

{S.n converges} C {Sn is bounded) C {Sn converges essentially)

can have probabilities [0, 0, 0], [0, 0, 1], [0, 1, 1] or [1, 1, 1].

We tu rn now to the three series criterion. First, let us assume tha t each 2 and let 2 t e rm has a finite variance aj *.n = E ~ - i a~ = a2(Sn)"

o o T h e o r e m 3.3. If E j : I ~rga < c>o then {Sn -E[Sn]} converges a.s. and in 2 s If sn ~ oc and the terms Xj are bounded by some constants 7j such

that maxj<nTj/sr, --+ 0., then {S.~} is essentially divergent (in particular

2 grows to a finite limit the a.s. Proof. Since ,~u has variance 2s~, when ~ convergence of {,~} is warranted (Theorem 2.5). Therefore there are constants C.~ such that S.~ C.~ ~ S. On tile other hand

E[(S.~,+k -- E[,S,,+k]- & + <&])~]- Z ~9 > 0 as . -+ oo; j--n+l

therefore {S~,- E[S~,]} is a Cauchy sequence in s 2, and has a limit S' in s

and, also, ill probability. From S,~-C,~ s S and S.~-E[S.~] k S', it follows tha t C , - E [ S . , ] converges to some constant C. Hence S~,-E[S.~] ~'~> S+C.

The second assertion is merely a res ta tement of the second half of The- orem 2.5, once it is observed tha t IXj X]. I < 27j. []

416 R. Vdlez

T h e o r e m 3.4. (Three series theorem) {S~} cow,verges a.s. if a~zd o~ty if

(z) Ej~P{IXjl > at,

(2) Ej~_~ ~(~5 ~,:~,z~),

(3) Ej~_I E[xj I{ixjl<~}]

co~tverge for some e > 0 (and theft they cow,verge for a~ty e > 0).

Proof. As pointed out in the proof of Theorem 2.6, {S~} converges a.s. if

and only if (1) is convergent and S~ ~) converges a.s.

When S![ ~) converges a.s. (2) must be convergent (by Theorem 3.3) and

then ,g~) - E[S(~ ~)] converges a.s. Hence E[,g~ ~)] converges and this asserts the convergence of (3).

On the other hand, if (2) and (3) converge, S~ ~ ) - E[S~ ~)] converges a.s.

and E[,g!~ )] is convergent. Thus ,g!~) converges a.s. []

Tile following well-known corollary, due to Lo6ve, can be easily dednced from the previous result (Chow 1978, pp. 117).

Corollary 3.1. If E j ~ E [ I x j l P , ] < ~ fo~-~om~ ~ q u ~ p j �9 (0,2] ~d E[Xj] = 0 whenever pj < 1, then {Sn} cow,verges a.s.

4 Examples

Tile following simple situation allows us to observe tile behaviour of {S~} for suitable choices of the parameters aj and pj. Especially they can be chosen such that S~ - C~ is a.s. convergent, for some constants C,~, while S,~ - E[S,~] is a.s. divergent.

Let {Xj} be independent random variables with distribution

P { X j - - aj} - -pj and P { X j - - - a j } - - 1 - p j ,

where aj > 0 and pj E [0, 1]. Tile symmetrized random variables are

~ { : 2 a j withprobability pj(1 pj) .~j X j - Xj with probability 1 - 2pj(1 - p j ) .


Clear ly EIXj] aj(2pj - 1), ~r2(Xj) 4a~pj(1 - pj) ~r2(&)/2 .

W h e n aj is a b o u n d e d sequence, Theorem 2.5 and 3.3 show t h a t 2 Y~j 1 aj pj (1 pj) < oc is a necessary and sufficient condi t ion for the a.s.

convergence of {+gn} and for tile a,s, c o n v e r g e n c e o f {St+ -- ~IS.p+I}, W h e n

Ej~_l aj (2pj 1) < c,x~, E[S+] is a convergent sequence, and {S+} is also a.s. convergent.

~> For aj l / j , it follows t h a t {S~} and {&+ - E[&+]} are a.s. convergent.

W i t h pj = 1/2 + 1/2j, E[S,] = E j ~ I 1/J 2 converges and {Sn} is a.s.

convergent.

W i t h pj = 3/4, E[S~] = E ~ ~ 1 1/2j and E[S~] ( l o g n ) / 2 --+ 7 / 2 (where 7 is the Euler 's constant) . Tlms {Sr~ - (log n ) / 2 - 7 /2} is a.s. convergent, bu t &+ a.,~ +oc .

o o t > F o r [ a j = j - 1 / 2 p j = l / 2 ] o r [ a j = l , p j = l/j], as E j l a]pJ(1 pj) = c~, {St,} is essential ly divergent: no cons tants Cr~ exist such tha t {Sr~ - Cr,} remains bounded. In tile first case, by symmet ry , l im supS,~ +or and l iminfS~, - ~ wi th probabi l i ty one. In tile second ease, tile te rms Xj

{Z.8. 1 are more and more fi 'equent, so cer ta in ly S n ~ oc.

W h e n aj is not bounded (aj ---+ 0 is obviously a necessary condi t ion for tile convergence of {S~,}), Theo rem 2.6 gives tha t {S~} is a.s. convergent if and only if, for some (or all) g > 0,

E p j ( 1 - p j ) < oo and E a ~ p j ( 1 - p j ) < oo. (4.1) {jlaa>c} {jlaa<c}

> For aj - j and pj - 1/j 2, the second sum has only a finite number of terms, while tile first one is convergent. Titus {S,~} converges a.s. (it will have a finite nmnber of non-vanishing terms, according to tile Borel-Cantel l i lemma) and some cons tants Ca can be found in order for {Sn - Ca} to be a.s. convergent.

Let us t ry C,, - EIS,~ 1. T h a t means we consider the r a n d o m terms

_ EIA~ ] f 2 j 2/j with probabi l i ty 1/j 2 - 2 / j with probabi l i ty 1 - 1 /9 .

418 R. Vdlez

o o c ,o S i n c e E j 2E[Yjl{i~;.i~}] = E j ~ (2/j)(1 1/j 2) = ~ , t h e t t n ' e e s e r i e s n y theorem shows that ~~j 1 J S n - - E[S,~] is not a.s. convergent. Hence

Y$

zls l = Z 2 = = j ~ 2 1 o g n + 2 7 n ( n + l ) / 2 j 1 j

is not a suitable choice. On tile other hand, for tile series of random terms

with probabil i ty 1/j 2

with probabil i ty 1 - 1/j 2,

P{IZj l > 1} 1/j 2, c~2(ZjI{IzjI<_I}) 0 and E[ZjI{Iz.I<_@ 0, so that r/, ~j=l Zj converges a.s., according to the three series theorem. Thus C~ =

n(n + 1)/2 is such that {S~ C.~] converges a.s.

R e f e r e n c e s

Chow, Y.S. and H. Teicher (1978). Probability Theory. Springer Verlag.

Chung, K.L. (1974). A Course in Probability Theory. Academic Press.

Doob, J.L. (1953). Stochastic Processes. John Wiley.

Fristedt, B. and L. Gray (1977). A Modern Approach to Probability Theory, Birkh~user.

Gihrnan, I.I. and A.V. Skorohod (1975). The Theory of Stochastic Processes IL Springer Verlag.

Lohve, M. (1977). Probability Theory. Springer Verlag.

Skorohod, A.V. (1986). Random Processes with Independent Increments. Kluwer Academic Publishers.

a new approach to series of independent random variables

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