convergence in probability refers only to the distribution of xn insteadif the sequence is defined on common probability spacethen we can consider the following mode of convergence that uses the joint distribution of the sequence of random variables definition almost sure convergence the sequence of random variables converges almost surely to random variable if for every lim sup |xk xe nk> almost sure convergence we denote the almost sure convergence as xn - note that in accordance with these definitions xn - is equivalent to supk> |xk - example (convergence in probability versus almost sure convergencesince the event {|xn xeis contained in {supk> |xk xe}we can conclude that almost sure convergence implies convergence in probability howeverthe converse is not true in general for instanceconsider the iid sequence with marginal distribution [xn [xn / clearlyxn - howeverfor and any we havep sup |xk [xn exn+ ek> [xn ex [xn+ ex (using independencem lim [xk elim mmk = = - - ** mn + it follows that [supk> |xk for any in other wordsit is not true that xn - lim another important type of convergence is useful when we are interested in estimating expectations or multidimensional integrals via monte carlo methodology definition convergence in distribution the sequence of random variables is said to converge in distribution to random variable with distribution function (xp[ xprovided thatlim [xn xf (xfor all such that lim (af (xnd convergence in distribution ( - we denote the convergence in distribution as either xn -xor xn -