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I've one question regarding the convergence of the sum of two stochastic processes. Let $(X^n_{t})_t \rightarrow (X_t)_t$ and $(Y^n_{t})_t \rightarrow (Y_t)_t$ for $n \to \infty$ where $\rightarrow$ denotes convergence in distribution in the uniform metric in $C[0,\infty)$. Assume that $X^n_t$ and $Y^n_t$ are independent as well as $X_t$ and $Y_t$ are independent. Does $(X^n_t+Y^n_t)_t \rightarrow (X_t+Y_t)_t$ hold? I know that the statement is true in fdd convergence but does it also hold if one regards the processes with the uniform metric?

Many thanks in advance!

pierreparis1
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  • what do you mean precisely by convergence in distribution in the uniform metric (or Skorokhod metric) ? – TheBridge Oct 25 '14 at 16:12
  • I mean the convergence in $C[0,\infty)$ which is used e.g. in Donsker's theorem. (the standard version of Donsker uses the uniform metric, but there's also one special case which needs the skorokhod metric (in $D[0,\infty)$).). – pierreparis1 Oct 27 '14 at 08:24

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