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Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces

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  • André Adler
  • Andrew Rosalsky
  • Andrej I. Volodin

Abstract

For a sequence of constants {a n,n≥1}, an array of rowwise independent and stochastically dominated random elements { V nj, j≥1, n≥1} in a real separable Rademacher type p (1≤p≤2) Banach space, and a sequence of positive integer-valued random variables {T n, n≥1}, a general weak law of large numbers of the form $$\sum {_{j = 1}^{T_n } } a_j (V_{nj} - c_{nj} )/b_{[\alpha _n ]} \xrightarrow{P}0$$ is established where {c nj, j≥1, n≥1}, α n → ∞, b n → ∞ are suitable sequences. Some related results are also presented. No assumption is made concerning the existence of expected values or absolute moments of the {V nj, j≥1, n≥1}. Illustrative examples include one wherein the strong law of large numbers fails.

Suggested Citation

  • André Adler & Andrew Rosalsky & Andrej I. Volodin, 1997. "Weak Laws with Random Indices for Arrays of Random Elements in Rademacher Type p Banach Spaces," Journal of Theoretical Probability, Springer, vol. 10(3), pages 605-623, July.
    • Handle: RePEc:spr:jotpro:v:10:y:1997:i:3:d:10.1023_a:1022645526197
    DOI: 10.1023/A:1022645526197
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    References listed on IDEAS

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    1. Wei, Duan & Taylor, R. L., 1978. "Convergence of weighted sums of tight random elements," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 282-294, June.
    2. Adler, André & Rosalsky, Andrew & Taylor, Robert L., 1991. "A weak law for normed weighted sums of random elements in rademacher type p banach spaces," Journal of Multivariate Analysis, Elsevier, vol. 37(2), pages 259-268, May.
    3. Andre Adler & Andrew Rosalsky & Robert L. Taylor, 1989. "Strong laws of large numbers for weighted sums of random elements in normed linear spaces," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 12, pages 1-23, January.
    4. André Adler & Andrew Rosalsky, 1991. "On the weak law of large numbers for normed weighted sums of I.I.D. random variables," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 14, pages 1-12, January.
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