In mathematics, Varadhan's lemma is a result from the large deviations theory named after S. R. Srinivasa Varadhan. The result gives information on the asymptotic distribution of a statistic "φ"("Z""ε") of a family of random variables "Z""ε" as "ε" becomes small in terms of a rate function for the variables.
Statement of the lemma.
Let "X" be a regular topological space; let ("Z""ε")"ε" ;0 be a family of random variables taking values in "X"; let "μ""ε" be the law (probability measure) of "Z""ε". Suppose that ("μ""ε")"ε" ;0 satisfies the large deviation principle with good rate function "I" : "X" → [0, +∞]. Let "ϕ"  : "X" → R be any continuous function. Suppose that at least one of the following two conditions holds true: either the tail condition
formula_1
where 1("E") denotes the indicator function of the event "E"; or, for some "γ"  ; 1, the moment condition
formula_2
Then
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