# Law of the iterated logarithm in C([0,1])

Let $$X_1, X_2, \ldots$$ be a sequence of independent identically distributed random variables with values in the space $$C([0,1])$$ of real continuous functions on $$[0,1$$]. Assume for simplicity that the $$X_i$$'s are centered, in the sense that $$E( X_i(t)) = 0$$ for $$t \in [0,1]$$). Are there any necessary and sufficient conditions (on the common distribution of the $$X_i$$'s) in order that $$\limsup_{n \to \infty} \frac {1}{\sqrt {2n \ln \ln (n)}} \, \| X_1 + \cdots +X_n \|_\infty < \infty$$ almost surely (where $$\| \cdot \|_\infty$$ is the sup norm)? What would then be the value of the limsup?

## 1 Answer

What you want is the so-called bounded law of the iterated logarithm (BLIL) in $$C[0,1]$$.

A necessary and sufficient condition for the BLIL in an arbitrary real separable Banach space $$B$$ was given by Ledoux and Talagrand '88 -- see e.g. Theorem A in the subsequent paper by Einmahl '93. This necessary and sufficient condition involves the boundedness of $$(S_n/\sqrt{n\ln\ln n})_{n\ge3}$$ in probability, which may be hard to verify in $$C[0,1]$$.

A sufficient condition (involving a certain, rather weak condition on the modulus of continuity of $$X_1(\cdot)$$) for the BLIL in $$C[0,1]$$ is implied by Theorem 2.1 in the earlier paper by Kuelbs '76.