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# A Note on Qualitative Conditions for the Strong Law of Large Numbers

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ANNALES

UNIVER8ITATIS MAEIAE C U E I E-S К Ł O D OWS К A LUBLIN - POLONIA

VOL. XVIII, 1 SECTIO A 1964

Z Katedry Statystyki Matematycznej Wydziału Mat. Fiz. Chem. UMCS Kierownik: prof. dr Mikołaj Olekiewicz

DOMINIK SZYNAL

A Note on Qualitative Conditions for the Strong Law ofLarge Numbers

Uwaga o jakościowych warunkach dla mocnego prawa wielkich liczb Замечание о качественных условиях для усиленного закона больших чисел

E. Eranckx in [1] considers a sequence of uniformly bounded random variables {Xn} and formulates his qualitative criterion: a necessary and sufficient condition for the strong law of large numbers (S. L. L. N) for such a sequence is the existence of a characteristic subsequence {Snklnk}

oi {S,Jn}, Sn — ^X, such that 3-1

(1)

(2)

lim nk+l

k-+oo Vlk 1,

Snk-ESnk a.s. Й

■ ■ —--- —> 0, where SKk — У A,.

«fc /Tj

In this note a case is considered when random variables are bounded but from one side, i.e. either Xn < I or Xn I, where Z is a finite number.

Without loss on generality either case can be reduced to the case of non negative random variables (Xn 0), for if I is negative and is a lower bound, we can by adding — I to random variables bring them to Xn 0, and if I is an upper bound we can by substracting I from random variables and multiplying by — 1 bring them to > 0.

Theorem 1. The 8. L. L. N. holds for a sequence of non negative random variables {Xn}(Xn 0 for all n) with bounded expectations EXn < L, if and only if there exists a characteristic subsequence (in Franckx's sense).

Proof. The necessity of condition is obvious from the fact that once a sequence {8n/n — E8nln} converges to zero a. s., any subsequence of this sequence converges to zero a.s.

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6 Dominik Szynal Condition is sufficient.

Since Xn 0, it follows that for nk < n <

(3) 8nk — ESnk+1 Sn — E8n < Snk+1 — ESnk.

From (2) we have: with probability 1 for sufficiently large k and an arbi­

trary positive constant e

—cnk-\- ESnk < Snk < ESnk-\- enk and also

< &Hk+l < + e^*+l •

Hence using (3) we obtain: with probability 1 for sufficiently large k - enk+1- (E8„k+1- E8„k) < Sn—ESn < (E8n/e+1—E8n/e) + enk+1, i.e.

(3'j ™»\ < E8nk+1 — E8nk + nk+l

n nk nk ’

and since EXn < L,

|\$n ESn\ (Hfc+i nk) L ft'k+i

--- <--- 1---e e,

n nk nk+1 nk

when k -> oo.

Thus we have: with probability 1 for sufficiently large n

\Sn-E8n\

n <e+ri, where e and g are arbitrary positive constants, i.e.

8n—E8n a.s.

—--- ”—> 0.

n

Corollary. For a sequence of non negative random variables {X„} with common finite expectation EXn — y the S. L. L. N. holds if and only if there is a characteristic subsequence (in Franckx's sense). Proof is immediate.

Theorem 2. The 8. L. L. N. holds for a sequence of non negative random variables if there exists such a subsequence of natural numbers nk] oo with

1 • 1

lim —— = c < oo that A'->oo 'W'A:

®nk _ pj ®nk a,s~ o

nk nk and E8nk+~E8nk

nk

The proof follows directly from (3') in the Theorem 1.

0.

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A note on qualitative conditions for the strong law of large numbers 7 KEEERENCES

[1] Franokx, E., La loi forte des grand nombres des variables uniformément bornées, Trab. de Estadist., 9 (1958), p. 111-115.

Streszczenie

W pracy tej dowodzi się, że warunkiem koniecznym i dostatecznym na to, aby ciąg zmiennych losowych nieujemnych o ograniczonych wartoś­

ciach oczekiwanych spełniał mocne prawo wielkich liczb, jest kryterium jakościowe Franckxa. Podaje się również jakościowe warunki dostateczne na to, aby ciąg zmiennych losowych nieujemnych spełniał mocne prawo wielkich liczb.

Резюме

В статье доказывается, что необходимым и достаточным условием для исполнения усиленного закона больших чисел в случае неотрица­

тельных случайных величин, имеющих ограниченные математические ожидания есть качественный критерий Франкса. Кроме того, даются достаточные качественные условия исполнения усиленного закона больших чисел в общем случае неотрицательных случайных величин.

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